Integrand size = 22, antiderivative size = 170 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 f F^{a+b c+b d x} (e+2 f x)}{b^4 d^4 \log ^4(F)}+\frac {2 F^{a+b c+b d x} \left (e^2+6 e f x+6 f^2 x^2\right )}{b^3 d^3 \log ^3(F)}-\frac {2 F^{a+b c+b d x} x \left (e^2+3 e f x+2 f^2 x^2\right )}{b^2 d^2 \log ^2(F)}+\frac {F^{a+b c+b d x} x^2 (e+f x)^2}{b d \log (F)} \] Output:
24*f^2*F^(b*d*x+b*c+a)/b^5/d^5/ln(F)^5-12*f*F^(b*d*x+b*c+a)*(2*f*x+e)/b^4/ d^4/ln(F)^4+2*F^(b*d*x+b*c+a)*(6*f^2*x^2+6*e*f*x+e^2)/b^3/d^3/ln(F)^3-2*F^ (b*d*x+b*c+a)*x*(2*f^2*x^2+3*e*f*x+e^2)/b^2/d^2/ln(F)^2+F^(b*d*x+b*c+a)*x^ 2*(f*x+e)^2/b/d/ln(F)
Time = 0.42 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.71 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {F^{a+b (c+d x)} \left (24 f^2-12 b d f (e+2 f x) \log (F)+2 b^2 d^2 \left (e^2+6 e f x+6 f^2 x^2\right ) \log ^2(F)-2 b^3 d^3 x \left (e^2+3 e f x+2 f^2 x^2\right ) \log ^3(F)+b^4 d^4 x^2 (e+f x)^2 \log ^4(F)\right )}{b^5 d^5 \log ^5(F)} \] Input:
Integrate[F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x]
Output:
(F^(a + b*(c + d*x))*(24*f^2 - 12*b*d*f*(e + 2*f*x)*Log[F] + 2*b^2*d^2*(e^ 2 + 6*e*f*x + 6*f^2*x^2)*Log[F]^2 - 2*b^3*d^3*x*(e^2 + 3*e*f*x + 2*f^2*x^2 )*Log[F]^3 + b^4*d^4*x^2*(e + f*x)^2*Log[F]^4))/(b^5*d^5*Log[F]^5)
Time = 1.45 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.93, 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.
| \(\displaystyle \int x^2 (e+f x)^2 F^{a+b (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (e^2 x^2 F^{a+b (c+d x)}+2 e f x^3 F^{a+b (c+d x)}+f^2 x^4 F^{a+b (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^4 F^{a+b c+b d x}}{b d \log (F)}\) |
Input:
Int[F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x]
Output:
(24*f^2*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) - (12*e*f*F^(a + b*c + b*d *x))/(b^4*d^4*Log[F]^4) - (24*f^2*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4 ) + (2*e^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (12*e*f*F^(a + b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) + (12*f^2*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*L og[F]^3) - (2*e^2*F^(a + b*c + b*d*x)*x)/(b^2*d^2*Log[F]^2) - (6*e*f*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (4*f^2*F^(a + b*c + b*d*x)*x^3)/( b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^2)/(b*d*Log[F]) + (2*e*f*F^ (a + b*c + b*d*x)*x^3)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^4)/(b*d*L og[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 = 0.07 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16
| method | result | size |
| gosper | \(\frac {\left (\ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}+2 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}+\ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}-2 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x +12 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} d^{2} e f x +2 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) b d \,f^{2} x -12 e f \ln \left (F \right ) b d +24 f^{2}\right ) F^{b d x +b c +a}}{\ln \left (F \right )^{5} b^{5} d^{5}}\) | \(197\) |
| risch | \(\frac {\left (\ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}+2 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}+\ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}-2 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x +12 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} d^{2} e f x +2 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) b d \,f^{2} x -12 e f \ln \left (F \right ) b d +24 f^{2}\right ) F^{b d x +b c +a}}{\ln \left (F \right )^{5} b^{5} d^{5}}\) | \(197\) |
| orering | \(\frac {\left (\ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}+2 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}+\ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}-2 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x +12 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} d^{2} e f x +2 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) b d \,f^{2} x -12 e f \ln \left (F \right ) b d +24 f^{2}\right ) F^{a +b \left (d x +c \right )}}{\ln \left (F \right )^{5} b^{5} d^{5}}\) | \(197\) |
| meijerg | \(-\frac {F^{b c +a} f^{2} \left (24-\frac {\left (5 b^{4} d^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-120 b d x \ln \left (F \right )+120\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{5}\right )}{\ln \left (F \right )^{5} b^{5} d^{5}}+\frac {2 F^{b c +a} f e \left (6-\frac {\left (-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 b d x \ln \left (F \right )+24\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{4}\right )}{b^{4} d^{4} \ln \left (F \right )^{4}}-\frac {F^{b c +a} e^{2} \left (2-\frac {\left (3 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-6 b d x \ln \left (F \right )+6\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{3}\right )}{b^{3} d^{3} \ln \left (F \right )^{3}}\) | \(217\) |
| norman | \(\frac {f^{2} x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{b d \ln \left (F \right )}+\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-6 e f \ln \left (F \right ) b d +12 f^{2}\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} d^{3}}+\frac {2 \left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-6 e f \ln \left (F \right ) b d +12 f^{2}\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} d^{5}}-\frac {2 \left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-6 e f \ln \left (F \right ) b d +12 f^{2}\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} d^{4}}+\frac {2 f \left (\ln \left (F \right ) b d e -2 f \right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} d^{2}}\) | \(233\) |
| parallelrisch | \(\frac {\ln \left (F \right )^{4} x^{4} F^{b d x +b c +a} b^{4} d^{4} f^{2}+2 \ln \left (F \right )^{4} x^{3} F^{b d x +b c +a} b^{4} d^{4} e f +\ln \left (F \right )^{4} x^{2} F^{b d x +b c +a} b^{4} d^{4} e^{2}-4 \ln \left (F \right )^{3} x^{3} F^{b d x +b c +a} b^{3} d^{3} f^{2}-6 \ln \left (F \right )^{3} x^{2} F^{b d x +b c +a} b^{3} d^{3} e f -2 \ln \left (F \right )^{3} x \,F^{b d x +b c +a} b^{3} d^{3} e^{2}+12 \ln \left (F \right )^{2} x^{2} F^{b d x +b c +a} b^{2} d^{2} f^{2}+12 \ln \left (F \right )^{2} x \,F^{b d x +b c +a} b^{2} d^{2} e f +2 \ln \left (F \right )^{2} F^{b d x +b c +a} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) x \,F^{b d x +b c +a} b d \,f^{2}-12 \ln \left (F \right ) F^{b d x +b c +a} b d e f +24 F^{b d x +b c +a} f^{2}}{\ln \left (F \right )^{5} b^{5} d^{5}}\) | \(318\) |
Input:
int(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x,method=_RETURNVERBOSE)
Output:
(ln(F)^4*b^4*d^4*f^2*x^4+2*ln(F)^4*b^4*d^4*e*f*x^3+ln(F)^4*b^4*d^4*e^2*x^2 -4*ln(F)^3*b^3*d^3*f^2*x^3-6*ln(F)^3*b^3*d^3*e*f*x^2-2*ln(F)^3*b^3*d^3*e^2 *x+12*ln(F)^2*b^2*d^2*f^2*x^2+12*ln(F)^2*b^2*d^2*e*f*x+2*ln(F)^2*b^2*d^2*e ^2-24*ln(F)*b*d*f^2*x-12*e*f*ln(F)*b*d+24*f^2)*F^(b*d*x+b*c+a)/ln(F)^5/b^5 /d^5
Time = 0.07 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.05 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {{\left ({\left (b^{4} d^{4} f^{2} x^{4} + 2 \, b^{4} d^{4} e f x^{3} + b^{4} d^{4} e^{2} x^{2}\right )} \log \left (F\right )^{4} - 2 \, {\left (2 \, b^{3} d^{3} f^{2} x^{3} + 3 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} + 2 \, {\left (6 \, b^{2} d^{2} f^{2} x^{2} + 6 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 24 \, f^{2} - 12 \, {\left (2 \, b d f^{2} x + b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{5} d^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="fricas")
Output:
((b^4*d^4*f^2*x^4 + 2*b^4*d^4*e*f*x^3 + b^4*d^4*e^2*x^2)*log(F)^4 - 2*(2*b ^3*d^3*f^2*x^3 + 3*b^3*d^3*e*f*x^2 + b^3*d^3*e^2*x)*log(F)^3 + 2*(6*b^2*d^ 2*f^2*x^2 + 6*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 + 24*f^2 - 12*(2*b*d*f ^2*x + b*d*e*f)*log(F))*F^(b*d*x + b*c + a)/(b^5*d^5*log(F)^5)
Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.53 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{4} d^{4} e^{2} x^{2} \log {\left (F \right )}^{4} + 2 b^{4} d^{4} e f x^{3} \log {\left (F \right )}^{4} + b^{4} d^{4} f^{2} x^{4} \log {\left (F \right )}^{4} - 2 b^{3} d^{3} e^{2} x \log {\left (F \right )}^{3} - 6 b^{3} d^{3} e f x^{2} \log {\left (F \right )}^{3} - 4 b^{3} d^{3} f^{2} x^{3} \log {\left (F \right )}^{3} + 2 b^{2} d^{2} e^{2} \log {\left (F \right )}^{2} + 12 b^{2} d^{2} e f x \log {\left (F \right )}^{2} + 12 b^{2} d^{2} f^{2} x^{2} \log {\left (F \right )}^{2} - 12 b d e f \log {\left (F \right )} - 24 b d f^{2} x \log {\left (F \right )} + 24 f^{2}\right )}{b^{5} d^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d^{5} \log {\left (F \right )}^{5} \neq 0 \\\frac {e^{2} x^{3}}{3} + \frac {e f x^{4}}{2} + \frac {f^{2} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(F**(a+b*(d*x+c))*x**2*(f*x+e)**2,x)
Output:
Piecewise((F**(a + b*(c + d*x))*(b**4*d**4*e**2*x**2*log(F)**4 + 2*b**4*d* *4*e*f*x**3*log(F)**4 + b**4*d**4*f**2*x**4*log(F)**4 - 2*b**3*d**3*e**2*x *log(F)**3 - 6*b**3*d**3*e*f*x**2*log(F)**3 - 4*b**3*d**3*f**2*x**3*log(F) **3 + 2*b**2*d**2*e**2*log(F)**2 + 12*b**2*d**2*e*f*x*log(F)**2 + 12*b**2* d**2*f**2*x**2*log(F)**2 - 12*b*d*e*f*log(F) - 24*b*d*f**2*x*log(F) + 24*f **2)/(b**5*d**5*log(F)**5), Ne(b**5*d**5*log(F)**5, 0)), (e**2*x**3/3 + e* f*x**4/2 + f**2*x**5/5, True))
Time = 0.06 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.54 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} e f}{b^{4} d^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{b c + a} b d x \log \left (F\right ) + 24 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{5} d^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="maxima")
Output:
(F^(b*c + a)*b^2*d^2*x^2*log(F)^2 - 2*F^(b*c + a)*b*d*x*log(F) + 2*F^(b*c + a))*F^(b*d*x)*e^2/(b^3*d^3*log(F)^3) + 2*(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*f/(b^4*d^4*log(F)^4) + (F^(b*c + a)*b^4*d^4*x^4*lo g(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*l og(F)^2 - 24*F^(b*c + a)*b*d*x*log(F) + 24*F^(b*c + a))*F^(b*d*x)*f^2/(b^5 *d^5*log(F)^5)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 6582, normalized size of antiderivative = 38.72 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\text {Too large to display} \] Input:
integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="giac")
Output:
-((2*(2*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 2*pi*b^4*d^4*f^2*x^4*log (abs(F))^3*sgn(F) - 2*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 2*pi*b^4*d^4*f^2* x^4*log(abs(F))^3 + 4*pi^3*b^4*d^4*e*f*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*d ^4*e*f*x^3*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*d^4*e*f*x^3*log(abs(F)) + 4*p i*b^4*d^4*e*f*x^3*log(abs(F))^3 + 2*pi^3*b^4*d^4*e^2*x^2*log(abs(F))*sgn(F ) - 2*pi*b^4*d^4*e^2*x^2*log(abs(F))^3*sgn(F) - 2*pi^3*b^4*d^4*e^2*x^2*log (abs(F)) + 2*pi*b^4*d^4*e^2*x^2*log(abs(F))^3 - 2*pi^3*b^3*d^3*f^2*x^3*sgn (F) + 6*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 2*pi^3*b^3*d^3*f^2*x^3 - 6*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 3*pi^3*b^3*d^3*e*f*x^2*sgn(F) + 9*pi *b^3*d^3*e*f*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*d^3*e*f*x^2 - 9*pi*b^3* d^3*e*f*x^2*log(abs(F))^2 - pi^3*b^3*d^3*e^2*x*sgn(F) + 3*pi*b^3*d^3*e^2*x *log(abs(F))^2*sgn(F) + pi^3*b^3*d^3*e^2*x - 3*pi*b^3*d^3*e^2*x*log(abs(F) )^2 - 12*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 12*pi*b^2*d^2*f^2*x^2*log (abs(F)) - 12*pi*b^2*d^2*e*f*x*log(abs(F))*sgn(F) + 12*pi*b^2*d^2*e*f*x*lo g(abs(F)) - 2*pi*b^2*d^2*e^2*log(abs(F))*sgn(F) + 2*pi*b^2*d^2*e^2*log(abs (F)) + 12*pi*b*d*f^2*x*sgn(F) - 12*pi*b*d*f^2*x + 6*pi*b*d*e*f*sgn(F) - 6* pi*b*d*e*f)*(pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs (F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d ^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*...
Time = 22.52 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.15 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {F^{a+b\,c+b\,d\,x}\,\left (b^4\,d^4\,e^2\,x^2\,{\ln \left (F\right )}^4+2\,b^4\,d^4\,e\,f\,x^3\,{\ln \left (F\right )}^4+b^4\,d^4\,f^2\,x^4\,{\ln \left (F\right )}^4-2\,b^3\,d^3\,e^2\,x\,{\ln \left (F\right )}^3-6\,b^3\,d^3\,e\,f\,x^2\,{\ln \left (F\right )}^3-4\,b^3\,d^3\,f^2\,x^3\,{\ln \left (F\right )}^3+2\,b^2\,d^2\,e^2\,{\ln \left (F\right )}^2+12\,b^2\,d^2\,e\,f\,x\,{\ln \left (F\right )}^2+12\,b^2\,d^2\,f^2\,x^2\,{\ln \left (F\right )}^2-12\,b\,d\,e\,f\,\ln \left (F\right )-24\,b\,d\,f^2\,x\,\ln \left (F\right )+24\,f^2\right )}{b^5\,d^5\,{\ln \left (F\right )}^5} \] Input:
int(F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x)
Output:
(F^(a + b*c + b*d*x)*(24*f^2 + 2*b^2*d^2*e^2*log(F)^2 - 24*b*d*f^2*x*log(F ) - 2*b^3*d^3*e^2*x*log(F)^3 + b^4*d^4*e^2*x^2*log(F)^4 + 12*b^2*d^2*f^2*x ^2*log(F)^2 - 4*b^3*d^3*f^2*x^3*log(F)^3 + b^4*d^4*f^2*x^4*log(F)^4 - 12*b *d*e*f*log(F) + 12*b^2*d^2*e*f*x*log(F)^2 - 6*b^3*d^3*e*f*x^2*log(F)^3 + 2 *b^4*d^4*e*f*x^3*log(F)^4))/(b^5*d^5*log(F)^5)
Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.15 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {f^{b d x +b c +a} \left (\mathrm {log}\left (f \right )^{4} b^{4} d^{4} e^{2} x^{2}+2 \mathrm {log}\left (f \right )^{4} b^{4} d^{4} e f \,x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} d^{4} f^{2} x^{4}-2 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} e^{2} x -6 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} e f \,x^{2}-4 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} f^{2} x^{3}+2 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e^{2}+12 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e f x +12 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} f^{2} x^{2}-12 \,\mathrm {log}\left (f \right ) b d e f -24 \,\mathrm {log}\left (f \right ) b d \,f^{2} x +24 f^{2}\right )}{\mathrm {log}\left (f \right )^{5} b^{5} d^{5}} \] Input:
int(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x)
Output:
(f**(a + b*c + b*d*x)*(log(f)**4*b**4*d**4*e**2*x**2 + 2*log(f)**4*b**4*d* *4*e*f*x**3 + log(f)**4*b**4*d**4*f**2*x**4 - 2*log(f)**3*b**3*d**3*e**2*x - 6*log(f)**3*b**3*d**3*e*f*x**2 - 4*log(f)**3*b**3*d**3*f**2*x**3 + 2*lo g(f)**2*b**2*d**2*e**2 + 12*log(f)**2*b**2*d**2*e*f*x + 12*log(f)**2*b**2* d**2*f**2*x**2 - 12*log(f)*b*d*e*f - 24*log(f)*b*d*f**2*x + 24*f**2))/(log (f)**5*b**5*d**5)