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Author: AlbertRich

Rubi 4 vs 5/Rubi 4 - 1.2.1.1 (a+b x+c x^2)^p.m

@@ -0,0 +1,143 @@
+(* ::Package:: *)
+
+(************************************************************************)
+(* This file was generated automatically by the Mathematica front end.  *)
+(* It contains Initialization cells from a Notebook file, which         *)
+(* typically will have the same name as this file except ending in      *)
+(* ".nb" instead of ".m".                                               *)
+(*                                                                      *)
+(* This file is intended to be loaded into the Mathematica kernel using *)
+(* the package loading commands Get or Needs.  Doing so is equivalent   *)
+(* to using the Evaluate Initialization Cells menu command in the front *)
+(* end.                                                                 *)
+(*                                                                      *)
+(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)
+(* automatically each time the parent Notebook file is saved in the     *)
+(* Mathematica front end.  Any changes you make to this file will be    *)
+(* overwritten.                                                         *)
+(************************************************************************)
+
+
+
+(* ::Code:: *)
+Int[(a_+b_.*x_+c_.*x_^2)^p_.,x_Symbol] :=
+  Int[Cancel[(b/2+c*x)^(2*p)/c^p],x] /;
+FreeQ[{a,b,c},x] && EqQ[b^2-4*a*c,0] && IntegerQ[p]
+
+
+(* ::Code:: *)
+Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  2*(a+b*x+c*x^2)^(p+1)/((2*p+1)*(b+2*c*x)) /;
+FreeQ[{a,b,c,p},x] && EqQ[b^2-4*a*c,0] && LtQ[p,-1]
+
+
+(* ::Code:: *)
+Int[1/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] :=
+  (b/2+c*x)/Sqrt[a+b*x+c*x^2] \[Star] Int[1/(b/2+c*x),x] /;
+FreeQ[{a,b,c},x] && EqQ[b^2-4*a*c,0]
+
+
+(* ::Code:: *)
+Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  (b+2*c*x)*(a+b*x+c*x^2)^p/(2*c*(2*p+1)) /;
+FreeQ[{a,b,c,p},x] && EqQ[b^2-4*a*c,0]
+
+
+(* ::Code:: *)
+Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  With[{q=Rt[b^2-4*a*c,2]}, 1/c^p \[Star] Int[Simp[b/2-q/2+c*x,x]^p*Simp[b/2+q/2+c*x,x]^p,x]] /;
+FreeQ[{a,b,c},x] && IntegerQ[p] && NeQ[a,0] && PerfectSquareQ[b^2-4*a*c]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  Int[ExpandIntegrand[(a+b*x+c*x^2)^p,x],x] /;
+FreeQ[{a,b,c},x] && IGtQ[p,0]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  (b+2*c*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) - 2*c*(2*p+3)/((p+1)*(b^2-4*a*c)) \[Star] Int[(a+b*x+c*x^2)^(p+1),x] /;
+FreeQ[{a,b,c},x] && ILtQ[p,-1]
+
+
+(* ::Code:: *)
+Int[1/(b_.*x_+c_.*x_^2),x_Symbol] :=
+  Log[x]/b - Log[RemoveContent[b+c*x,x]]/b /;
+FreeQ[{b,c},x]
+
+
+(* ::Code:: *)
+Int[1/(a_+b_.*x_+c_.*x_^2),x_Symbol] :=
+  With[{q=1-4*Simplify[a*c/b^2]}, -2/b \[Star] Subst[Int[1/(q-x^2),x],x,1+2*c*x/b] /;
+ RationalQ[q] && (EqQ[q^2,1] || Not[RationalQ[b^2-4*a*c]])] /;
+FreeQ[{a,b,c},x]
+
+
+(* ::Code:: *)
+Int[1/(a_+b_.*x_+c_.*x_^2),x_Symbol] :=
+  -2 \[Star] Subst[Int[1/Simp[b^2-4*a*c-x^2,x],x],x,b+2*c*x] /;
+FreeQ[{a,b,c},x]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  (b+2*c*x)*(a+b*x+c*x^2)^p/(2*c*(2*p+1)) - p*(b^2-4*a*c)/(2*c*(2*p+1)) \[Star] Int[(a+b*x+c*x^2)^(p-1),x] /;
+FreeQ[{a,b,c},x] && GtQ[p,0] && (IntegerQ[4*p] || IntegerQ[3*p])
+
+
+(* ::Code:: *)
+Int[1/(a_.+b_.*x_+c_.*x_^2)^(3/2),x_Symbol] :=
+  -2*(b+2*c*x)/((b^2-4*a*c)*Sqrt[a+b*x+c*x^2]) /;
+FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  (b+2*c*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) - 2*c*(2*p+3)/((p+1)*(b^2-4*a*c)) \[Star] Int[(a+b*x+c*x^2)^(p+1),x] /;
+FreeQ[{a,b,c},x] && LtQ[p,-1] && (IntegerQ[4*p] || IntegerQ[3*p])
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  1/(2*c*(-4*c/(b^2-4*a*c))^p) \[Star] Subst[Int[Simp[1-x^2/(b^2-4*a*c),x]^p,x],x,b+2*c*x] /;
+FreeQ[{a,b,c,p},x] && GtQ[4*a-b^2/c,0]
+
+
+(* ::Code:: *)
+Int[1/Sqrt[b_.*x_+c_.*x_^2],x_Symbol] :=
+  2 \[Star] Subst[Int[1/(1-c*x^2),x],x,x/Sqrt[b*x+c*x^2]] /;
+FreeQ[{b,c},x]
+
+
+(* ::Code:: *)
+Int[1/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] :=
+  2 \[Star] Subst[Int[1/(4*c-x^2),x],x,(b+2*c*x)/Sqrt[a+b*x+c*x^2]] /;
+FreeQ[{a,b,c},x]
+
+
+(* ::Code:: *)
+Int[(b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  (b*x+c*x^2)^p/(-c*(b*x+c*x^2)/(b^2))^p \[Star] Int[(-c*x/b-c^2*x^2/b^2)^p,x] /;
+FreeQ[{b,c},x] && (IntegerQ[4*p] || IntegerQ[3*p])
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  4*Sqrt[(b+2*c*x)^2]/(b+2*c*x) \[Star] Subst[Int[x^(4*(p+1)-1)/Sqrt[b^2-4*a*c+4*c*x^4],x],x,(a+b*x+c*x^2)^(1/4)] /;
+FreeQ[{a,b,c},x] && IntegerQ[4*p]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  3*Sqrt[(b+2*c*x)^2]/(b+2*c*x) \[Star] Subst[Int[x^(3*(p+1)-1)/Sqrt[b^2-4*a*c+4*c*x^3],x],x,(a+b*x+c*x^2)^(1/3)] /;
+FreeQ[{a,b,c},x] && IntegerQ[3*p]
+
+
+(* ::Code:: *)
+Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] :=
+  With[{q=Rt[b^2-4*a*c,2]}, -(a+b*x+c*x^2)^(p+1)/(q*(p+1)*((q-b-2*c*x)/(2*q))^(p+1))*Hypergeometric2F1[-p,p+1,p+2,(b+q+2*c*x)/(2*q)]] /;
+FreeQ[{a,b,c,p},x] && Not[IntegerQ[4*p]] && Not[IntegerQ[3*p]]
+
+
+