Integrand size = 22, antiderivative size = 219 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=-\frac {120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac {24 f F^{a+b c+b d x} (2 e+5 f x)}{b^5 d^5 \log ^5(F)}-\frac {6 F^{a+b c+b d x} \left (e^2+8 e f x+10 f^2 x^2\right )}{b^4 d^4 \log ^4(F)}+\frac {2 F^{a+b c+b d x} x \left (3 e^2+12 e f x+10 f^2 x^2\right )}{b^3 d^3 \log ^3(F)}-\frac {F^{a+b c+b d x} x^2 \left (3 e^2+8 e f x+5 f^2 x^2\right )}{b^2 d^2 \log ^2(F)}+\frac {F^{a+b c+b d x} x^3 (e+f x)^2}{b d \log (F)} \] Output:
-120*f^2*F^(b*d*x+b*c+a)/b^6/d^6/ln(F)^6+24*f*F^(b*d*x+b*c+a)*(5*f*x+2*e)/ b^5/d^5/ln(F)^5-6*F^(b*d*x+b*c+a)*(10*f^2*x^2+8*e*f*x+e^2)/b^4/d^4/ln(F)^4 +2*F^(b*d*x+b*c+a)*x*(10*f^2*x^2+12*e*f*x+3*e^2)/b^3/d^3/ln(F)^3-F^(b*d*x+ b*c+a)*x^2*(5*f^2*x^2+8*e*f*x+3*e^2)/b^2/d^2/ln(F)^2+F^(b*d*x+b*c+a)*x^3*( f*x+e)^2/b/d/ln(F)
Time = 0.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.73 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\frac {F^{a+b (c+d x)} \left (-120 f^2+24 b d f (2 e+5 f x) \log (F)-6 b^2 d^2 \left (e^2+8 e f x+10 f^2 x^2\right ) \log ^2(F)+2 b^3 d^3 x \left (3 e^2+12 e f x+10 f^2 x^2\right ) \log ^3(F)-b^4 d^4 x^2 \left (3 e^2+8 e f x+5 f^2 x^2\right ) \log ^4(F)+b^5 d^5 x^3 (e+f x)^2 \log ^5(F)\right )}{b^6 d^6 \log ^6(F)} \] Input:
Integrate[F^(a + b*(c + d*x))*x^3*(e + f*x)^2,x]
Output:
(F^(a + b*(c + d*x))*(-120*f^2 + 24*b*d*f*(2*e + 5*f*x)*Log[F] - 6*b^2*d^2 *(e^2 + 8*e*f*x + 10*f^2*x^2)*Log[F]^2 + 2*b^3*d^3*x*(3*e^2 + 12*e*f*x + 1 0*f^2*x^2)*Log[F]^3 - b^4*d^4*x^2*(3*e^2 + 8*e*f*x + 5*f^2*x^2)*Log[F]^4 + b^5*d^5*x^3*(e + f*x)^2*Log[F]^5))/(b^6*d^6*Log[F]^6)
Time = 1.63 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2626, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 (e+f x)^2 F^{a+b (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (e^2 x^3 F^{a+b (c+d x)}+2 e f x^4 F^{a+b (c+d x)}+f^2 x^5 F^{a+b (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac {48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac {120 f^2 x F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {48 e f x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {60 f^2 x^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {6 e^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {24 e f x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 x^3 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {8 e f x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 x^4 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^4 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^5 F^{a+b c+b d x}}{b d \log (F)}\) |
Input:
Int[F^(a + b*(c + d*x))*x^3*(e + f*x)^2,x]
Output:
(-120*f^2*F^(a + b*c + b*d*x))/(b^6*d^6*Log[F]^6) + (48*e*f*F^(a + b*c + b *d*x))/(b^5*d^5*Log[F]^5) + (120*f^2*F^(a + b*c + b*d*x)*x)/(b^5*d^5*Log[F ]^5) - (6*e^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (48*e*f*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) - (60*f^2*F^(a + b*c + b*d*x)*x^2)/(b^4*d^ 4*Log[F]^4) + (6*e^2*F^(a + b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) + (24*e*f*F ^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) + (20*f^2*F^(a + b*c + b*d*x)*x ^3)/(b^3*d^3*Log[F]^3) - (3*e^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2 ) - (8*e*f*F^(a + b*c + b*d*x)*x^3)/(b^2*d^2*Log[F]^2) - (5*f^2*F^(a + b*c + b*d*x)*x^4)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^3)/(b*d*Log [F]) + (2*e*f*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F]) + (f^2*F^(a + b*c + b* d*x)*x^5)/(b*d*Log[F])
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.08 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.14
| method | result | size |
| gosper | \(\frac {\left (f^{2} x^{5} \ln \left (F \right )^{5} b^{5} d^{5}+2 \ln \left (F \right )^{5} b^{5} d^{5} e f \,x^{4}+\ln \left (F \right )^{5} b^{5} d^{5} e^{2} x^{3}-5 \ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}-8 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}-3 \ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}+20 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+24 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+6 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -60 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-48 \ln \left (F \right )^{2} b^{2} d^{2} e f x -6 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}+120 \ln \left (F \right ) b d \,f^{2} x +48 e f \ln \left (F \right ) b d -120 f^{2}\right ) F^{b d x +b c +a}}{\ln \left (F \right )^{6} b^{6} d^{6}}\) | \(250\) |
| risch | \(\frac {\left (f^{2} x^{5} \ln \left (F \right )^{5} b^{5} d^{5}+2 \ln \left (F \right )^{5} b^{5} d^{5} e f \,x^{4}+\ln \left (F \right )^{5} b^{5} d^{5} e^{2} x^{3}-5 \ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}-8 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}-3 \ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}+20 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+24 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+6 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -60 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-48 \ln \left (F \right )^{2} b^{2} d^{2} e f x -6 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}+120 \ln \left (F \right ) b d \,f^{2} x +48 e f \ln \left (F \right ) b d -120 f^{2}\right ) F^{b d x +b c +a}}{\ln \left (F \right )^{6} b^{6} d^{6}}\) | \(250\) |
| orering | \(\frac {\left (f^{2} x^{5} \ln \left (F \right )^{5} b^{5} d^{5}+2 \ln \left (F \right )^{5} b^{5} d^{5} e f \,x^{4}+\ln \left (F \right )^{5} b^{5} d^{5} e^{2} x^{3}-5 \ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}-8 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}-3 \ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}+20 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+24 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+6 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -60 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-48 \ln \left (F \right )^{2} b^{2} d^{2} e f x -6 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}+120 \ln \left (F \right ) b d \,f^{2} x +48 e f \ln \left (F \right ) b d -120 f^{2}\right ) F^{a +b \left (d x +c \right )}}{\ln \left (F \right )^{6} b^{6} d^{6}}\) | \(250\) |
| meijerg | \(\frac {F^{b c +a} f^{2} \left (120-\frac {\left (-6 b^{5} d^{5} x^{5} \ln \left (F \right )^{5}+30 b^{4} d^{4} x^{4} \ln \left (F \right )^{4}-120 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+360 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-720 b d x \ln \left (F \right )+720\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{6}\right )}{\ln \left (F \right )^{6} b^{6} d^{6}}-\frac {2 F^{b c +a} f e \left (24-\frac {\left (5 b^{4} d^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-120 b d x \ln \left (F \right )+120\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{5}\right )}{b^{5} d^{5} \ln \left (F \right )^{5}}+\frac {F^{b c +a} e^{2} \left (6-\frac {\left (-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 b d x \ln \left (F \right )+24\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{4}\right )}{b^{4} d^{4} \ln \left (F \right )^{4}}\) | \(260\) |
| norman | \(\frac {f^{2} x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{b d \ln \left (F \right )}+\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-8 e f \ln \left (F \right ) b d +20 f^{2}\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} d^{3}}+\frac {f \left (2 \ln \left (F \right ) b d e -5 f \right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} d^{2}}-\frac {6 \left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-8 e f \ln \left (F \right ) b d +20 f^{2}\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6} d^{6}}+\frac {6 \left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-8 e f \ln \left (F \right ) b d +20 f^{2}\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} d^{5}}-\frac {3 \left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-8 e f \ln \left (F \right ) b d +20 f^{2}\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} d^{4}}\) | \(289\) |
| parallelrisch | \(\frac {x^{5} F^{b d x +b c +a} f^{2} \ln \left (F \right )^{5} b^{5} d^{5}+2 \ln \left (F \right )^{5} x^{4} F^{b d x +b c +a} b^{5} d^{5} e f +\ln \left (F \right )^{5} x^{3} F^{b d x +b c +a} b^{5} d^{5} e^{2}-5 \ln \left (F \right )^{4} x^{4} F^{b d x +b c +a} b^{4} d^{4} f^{2}-8 \ln \left (F \right )^{4} x^{3} F^{b d x +b c +a} b^{4} d^{4} e f -3 \ln \left (F \right )^{4} x^{2} F^{b d x +b c +a} b^{4} d^{4} e^{2}+20 \ln \left (F \right )^{3} x^{3} F^{b d x +b c +a} b^{3} d^{3} f^{2}+24 \ln \left (F \right )^{3} x^{2} F^{b d x +b c +a} b^{3} d^{3} e f +6 \ln \left (F \right )^{3} x \,F^{b d x +b c +a} b^{3} d^{3} e^{2}-60 \ln \left (F \right )^{2} x^{2} F^{b d x +b c +a} b^{2} d^{2} f^{2}-48 \ln \left (F \right )^{2} x \,F^{b d x +b c +a} b^{2} d^{2} e f -6 \ln \left (F \right )^{2} F^{b d x +b c +a} b^{2} d^{2} e^{2}+120 \ln \left (F \right ) x \,F^{b d x +b c +a} b d \,f^{2}+48 \ln \left (F \right ) F^{b d x +b c +a} b d e f -120 F^{b d x +b c +a} f^{2}}{\ln \left (F \right )^{6} b^{6} d^{6}}\) | \(404\) |
Input:
int(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
(f^2*x^5*ln(F)^5*b^5*d^5+2*ln(F)^5*b^5*d^5*e*f*x^4+ln(F)^5*b^5*d^5*e^2*x^3 -5*ln(F)^4*b^4*d^4*f^2*x^4-8*ln(F)^4*b^4*d^4*e*f*x^3-3*ln(F)^4*b^4*d^4*e^2 *x^2+20*ln(F)^3*b^3*d^3*f^2*x^3+24*ln(F)^3*b^3*d^3*e*f*x^2+6*ln(F)^3*b^3*d ^3*e^2*x-60*ln(F)^2*b^2*d^2*f^2*x^2-48*ln(F)^2*b^2*d^2*e*f*x-6*ln(F)^2*b^2 *d^2*e^2+120*ln(F)*b*d*f^2*x+48*e*f*ln(F)*b*d-120*f^2)*F^(b*d*x+b*c+a)/ln( F)^6/b^6/d^6
Time = 0.07 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.04 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\frac {{\left ({\left (b^{5} d^{5} f^{2} x^{5} + 2 \, b^{5} d^{5} e f x^{4} + b^{5} d^{5} e^{2} x^{3}\right )} \log \left (F\right )^{5} - {\left (5 \, b^{4} d^{4} f^{2} x^{4} + 8 \, b^{4} d^{4} e f x^{3} + 3 \, b^{4} d^{4} e^{2} x^{2}\right )} \log \left (F\right )^{4} + 2 \, {\left (10 \, b^{3} d^{3} f^{2} x^{3} + 12 \, b^{3} d^{3} e f x^{2} + 3 \, b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} - 6 \, {\left (10 \, b^{2} d^{2} f^{2} x^{2} + 8 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 120 \, f^{2} + 24 \, {\left (5 \, b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{6} d^{6} \log \left (F\right )^{6}} \] Input:
integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="fricas")
Output:
((b^5*d^5*f^2*x^5 + 2*b^5*d^5*e*f*x^4 + b^5*d^5*e^2*x^3)*log(F)^5 - (5*b^4 *d^4*f^2*x^4 + 8*b^4*d^4*e*f*x^3 + 3*b^4*d^4*e^2*x^2)*log(F)^4 + 2*(10*b^3 *d^3*f^2*x^3 + 12*b^3*d^3*e*f*x^2 + 3*b^3*d^3*e^2*x)*log(F)^3 - 6*(10*b^2* d^2*f^2*x^2 + 8*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 - 120*f^2 + 24*(5*b* d*f^2*x + 2*b*d*e*f)*log(F))*F^(b*d*x + b*c + a)/(b^6*d^6*log(F)^6)
Time = 0.11 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.47 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{5} d^{5} e^{2} x^{3} \log {\left (F \right )}^{5} + 2 b^{5} d^{5} e f x^{4} \log {\left (F \right )}^{5} + b^{5} d^{5} f^{2} x^{5} \log {\left (F \right )}^{5} - 3 b^{4} d^{4} e^{2} x^{2} \log {\left (F \right )}^{4} - 8 b^{4} d^{4} e f x^{3} \log {\left (F \right )}^{4} - 5 b^{4} d^{4} f^{2} x^{4} \log {\left (F \right )}^{4} + 6 b^{3} d^{3} e^{2} x \log {\left (F \right )}^{3} + 24 b^{3} d^{3} e f x^{2} \log {\left (F \right )}^{3} + 20 b^{3} d^{3} f^{2} x^{3} \log {\left (F \right )}^{3} - 6 b^{2} d^{2} e^{2} \log {\left (F \right )}^{2} - 48 b^{2} d^{2} e f x \log {\left (F \right )}^{2} - 60 b^{2} d^{2} f^{2} x^{2} \log {\left (F \right )}^{2} + 48 b d e f \log {\left (F \right )} + 120 b d f^{2} x \log {\left (F \right )} - 120 f^{2}\right )}{b^{6} d^{6} \log {\left (F \right )}^{6}} & \text {for}\: b^{6} d^{6} \log {\left (F \right )}^{6} \neq 0 \\\frac {e^{2} x^{4}}{4} + \frac {2 e f x^{5}}{5} + \frac {f^{2} x^{6}}{6} & \text {otherwise} \end {cases} \] Input:
integrate(F**(a+b*(d*x+c))*x**3*(f*x+e)**2,x)
Output:
Piecewise((F**(a + b*(c + d*x))*(b**5*d**5*e**2*x**3*log(F)**5 + 2*b**5*d* *5*e*f*x**4*log(F)**5 + b**5*d**5*f**2*x**5*log(F)**5 - 3*b**4*d**4*e**2*x **2*log(F)**4 - 8*b**4*d**4*e*f*x**3*log(F)**4 - 5*b**4*d**4*f**2*x**4*log (F)**4 + 6*b**3*d**3*e**2*x*log(F)**3 + 24*b**3*d**3*e*f*x**2*log(F)**3 + 20*b**3*d**3*f**2*x**3*log(F)**3 - 6*b**2*d**2*e**2*log(F)**2 - 48*b**2*d* *2*e*f*x*log(F)**2 - 60*b**2*d**2*f**2*x**2*log(F)**2 + 48*b*d*e*f*log(F) + 120*b*d*f**2*x*log(F) - 120*f**2)/(b**6*d**6*log(F)**6), Ne(b**6*d**6*lo g(F)**6, 0)), (e**2*x**4/4 + 2*e*f*x**5/5 + f**2*x**6/6, True))
Time = 0.07 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.50 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\frac {{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{4} d^{4} \log \left (F\right )^{4}} + \frac {2 \, {\left (F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{b c + a} b d x \log \left (F\right ) + 24 \, F^{b c + a}\right )} F^{b d x} e f}{b^{5} d^{5} \log \left (F\right )^{5}} + \frac {{\left (F^{b c + a} b^{5} d^{5} x^{5} \log \left (F\right )^{5} - 5 \, F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} + 20 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 60 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 120 \, F^{b c + a} b d x \log \left (F\right ) - 120 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{6} d^{6} \log \left (F\right )^{6}} \] Input:
integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="maxima")
Output:
(F^(b*c + a)*b^3*d^3*x^3*log(F)^3 - 3*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 6 *F^(b*c + a)*b*d*x*log(F) - 6*F^(b*c + a))*F^(b*d*x)*e^2/(b^4*d^4*log(F)^4 ) + 2*(F^(b*c + a)*b^4*d^4*x^4*log(F)^4 - 4*F^(b*c + a)*b^3*d^3*x^3*log(F) ^3 + 12*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 - 24*F^(b*c + a)*b*d*x*log(F) + 2 4*F^(b*c + a))*F^(b*d*x)*e*f/(b^5*d^5*log(F)^5) + (F^(b*c + a)*b^5*d^5*x^5 *log(F)^5 - 5*F^(b*c + a)*b^4*d^4*x^4*log(F)^4 + 20*F^(b*c + a)*b^3*d^3*x^ 3*log(F)^3 - 60*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 120*F^(b*c + a)*b*d*x*l og(F) - 120*F^(b*c + a))*F^(b*d*x)*f^2/(b^6*d^6*log(F)^6)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 9584, normalized size of antiderivative = 43.76 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\text {Too large to display} \] Input:
integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="giac")
Output:
-(((5*pi^4*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*f^2*x^5*lo g(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*f^2*x^5*log(abs(F)) + 10*pi^2*b^5*d^5* f^2*x^5*log(abs(F))^3 - 2*b^5*d^5*f^2*x^5*log(abs(F))^5 + 10*pi^4*b^5*d^5* e*f*x^4*log(abs(F))*sgn(F) - 20*pi^2*b^5*d^5*e*f*x^4*log(abs(F))^3*sgn(F) - 10*pi^4*b^5*d^5*e*f*x^4*log(abs(F)) + 20*pi^2*b^5*d^5*e*f*x^4*log(abs(F) )^3 - 4*b^5*d^5*e*f*x^4*log(abs(F))^5 + 5*pi^4*b^5*d^5*e^2*x^3*log(abs(F)) *sgn(F) - 10*pi^2*b^5*d^5*e^2*x^3*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*e^ 2*x^3*log(abs(F)) + 10*pi^2*b^5*d^5*e^2*x^3*log(abs(F))^3 - 2*b^5*d^5*e^2* x^3*log(abs(F))^5 - 5*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 30*pi^2*b^4*d^4*f^2*x^ 4*log(abs(F))^2*sgn(F) + 5*pi^4*b^4*d^4*f^2*x^4 - 30*pi^2*b^4*d^4*f^2*x^4* log(abs(F))^2 + 10*b^4*d^4*f^2*x^4*log(abs(F))^4 - 8*pi^4*b^4*d^4*e*f*x^3* sgn(F) + 48*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2*sgn(F) + 8*pi^4*b^4*d^4*e*f *x^3 - 48*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2 + 16*b^4*d^4*e*f*x^3*log(abs( F))^4 - 3*pi^4*b^4*d^4*e^2*x^2*sgn(F) + 18*pi^2*b^4*d^4*e^2*x^2*log(abs(F) )^2*sgn(F) + 3*pi^4*b^4*d^4*e^2*x^2 - 18*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^ 2 + 6*b^4*d^4*e^2*x^2*log(abs(F))^4 - 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F))* sgn(F) + 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 40*b^3*d^3*f^2*x^3*log(abs( F))^3 - 72*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) + 72*pi^2*b^3*d^3*e*f*x ^2*log(abs(F)) - 48*b^3*d^3*e*f*x^2*log(abs(F))^3 - 18*pi^2*b^3*d^3*e^2*x* log(abs(F))*sgn(F) + 18*pi^2*b^3*d^3*e^2*x*log(abs(F)) - 12*b^3*d^3*e^2...
Time = 22.65 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.14 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\frac {F^{a+b\,c+b\,d\,x}\,\left (b^5\,d^5\,e^2\,x^3\,{\ln \left (F\right )}^5+2\,b^5\,d^5\,e\,f\,x^4\,{\ln \left (F\right )}^5+b^5\,d^5\,f^2\,x^5\,{\ln \left (F\right )}^5-3\,b^4\,d^4\,e^2\,x^2\,{\ln \left (F\right )}^4-8\,b^4\,d^4\,e\,f\,x^3\,{\ln \left (F\right )}^4-5\,b^4\,d^4\,f^2\,x^4\,{\ln \left (F\right )}^4+6\,b^3\,d^3\,e^2\,x\,{\ln \left (F\right )}^3+24\,b^3\,d^3\,e\,f\,x^2\,{\ln \left (F\right )}^3+20\,b^3\,d^3\,f^2\,x^3\,{\ln \left (F\right )}^3-6\,b^2\,d^2\,e^2\,{\ln \left (F\right )}^2-48\,b^2\,d^2\,e\,f\,x\,{\ln \left (F\right )}^2-60\,b^2\,d^2\,f^2\,x^2\,{\ln \left (F\right )}^2+48\,b\,d\,e\,f\,\ln \left (F\right )+120\,b\,d\,f^2\,x\,\ln \left (F\right )-120\,f^2\right )}{b^6\,d^6\,{\ln \left (F\right )}^6} \] Input:
int(F^(a + b*(c + d*x))*x^3*(e + f*x)^2,x)
Output:
(F^(a + b*c + b*d*x)*(120*b*d*f^2*x*log(F) - 6*b^2*d^2*e^2*log(F)^2 - 120* f^2 + 6*b^3*d^3*e^2*x*log(F)^3 - 3*b^4*d^4*e^2*x^2*log(F)^4 + b^5*d^5*e^2* x^3*log(F)^5 - 60*b^2*d^2*f^2*x^2*log(F)^2 + 20*b^3*d^3*f^2*x^3*log(F)^3 - 5*b^4*d^4*f^2*x^4*log(F)^4 + b^5*d^5*f^2*x^5*log(F)^5 + 48*b*d*e*f*log(F) - 48*b^2*d^2*e*f*x*log(F)^2 + 24*b^3*d^3*e*f*x^2*log(F)^3 - 8*b^4*d^4*e*f *x^3*log(F)^4 + 2*b^5*d^5*e*f*x^4*log(F)^5))/(b^6*d^6*log(F)^6)
Time = 0.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.14 \[ \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx=\frac {f^{b d x +b c +a} \left (\mathrm {log}\left (f \right )^{5} b^{5} d^{5} e^{2} x^{3}+2 \mathrm {log}\left (f \right )^{5} b^{5} d^{5} e f \,x^{4}+\mathrm {log}\left (f \right )^{5} b^{5} d^{5} f^{2} x^{5}-3 \mathrm {log}\left (f \right )^{4} b^{4} d^{4} e^{2} x^{2}-8 \mathrm {log}\left (f \right )^{4} b^{4} d^{4} e f \,x^{3}-5 \mathrm {log}\left (f \right )^{4} b^{4} d^{4} f^{2} x^{4}+6 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} e^{2} x +24 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} e f \,x^{2}+20 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e^{2}-48 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e f x -60 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} f^{2} x^{2}+48 \,\mathrm {log}\left (f \right ) b d e f +120 \,\mathrm {log}\left (f \right ) b d \,f^{2} x -120 f^{2}\right )}{\mathrm {log}\left (f \right )^{6} b^{6} d^{6}} \] Input:
int(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x)
Output:
(f**(a + b*c + b*d*x)*(log(f)**5*b**5*d**5*e**2*x**3 + 2*log(f)**5*b**5*d* *5*e*f*x**4 + log(f)**5*b**5*d**5*f**2*x**5 - 3*log(f)**4*b**4*d**4*e**2*x **2 - 8*log(f)**4*b**4*d**4*e*f*x**3 - 5*log(f)**4*b**4*d**4*f**2*x**4 + 6 *log(f)**3*b**3*d**3*e**2*x + 24*log(f)**3*b**3*d**3*e*f*x**2 + 20*log(f)* *3*b**3*d**3*f**2*x**3 - 6*log(f)**2*b**2*d**2*e**2 - 48*log(f)**2*b**2*d* *2*e*f*x - 60*log(f)**2*b**2*d**2*f**2*x**2 + 48*log(f)*b*d*e*f + 120*log( f)*b*d*f**2*x - 120*f**2))/(log(f)**6*b**6*d**6)