Integrand size = 22, antiderivative size = 177 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {24 f^2 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {12 f F^{c (a+b x)} (e+2 f x)}{b^4 c^4 \log ^4(F)}+\frac {2 F^{c (a+b x)} \left (e^2+2 d f+6 e f x+6 f^2 x^2\right )}{b^3 c^3 \log ^3(F)}-\frac {2 F^{c (a+b x)} \left (d e+\left (e^2+2 d f\right ) x+3 e f x^2+2 f^2 x^3\right )}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} \left (d+e x+f x^2\right )^2}{b c \log (F)} \] Output:
24*f^2*F^(c*(b*x+a))/b^5/c^5/ln(F)^5-12*f*F^(c*(b*x+a))*(2*f*x+e)/b^4/c^4/ ln(F)^4+2*F^(c*(b*x+a))*(6*f^2*x^2+6*e*f*x+2*d*f+e^2)/b^3/c^3/ln(F)^3-2*F^ (c*(b*x+a))*(d*e+(2*d*f+e^2)*x+3*e*f*x^2+2*f^2*x^3)/b^2/c^2/ln(F)^2+F^(c*( b*x+a))*(f*x^2+e*x+d)^2/b/c/ln(F)
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.68 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {F^{c (a+b x)} \left (24 f^2-12 b c f (e+2 f x) \log (F)+2 b^2 c^2 \left (e^2+6 e f x+2 f \left (d+3 f x^2\right )\right ) \log ^2(F)-2 b^3 c^3 (e+2 f x) (d+x (e+f x)) \log ^3(F)+b^4 c^4 (d+x (e+f x))^2 \log ^4(F)\right )}{b^5 c^5 \log ^5(F)} \] Input:
Integrate[F^(c*(a + b*x))*(d + e*x + f*x^2)^2,x]
Output:
(F^(c*(a + b*x))*(24*f^2 - 12*b*c*f*(e + 2*f*x)*Log[F] + 2*b^2*c^2*(e^2 + 6*e*f*x + 2*f*(d + 3*f*x^2))*Log[F]^2 - 2*b^3*c^3*(e + 2*f*x)*(d + x*(e + f*x))*Log[F]^3 + b^4*c^4*(d + x*(e + f*x))^2*Log[F]^4))/(b^5*c^5*Log[F]^5)
Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(177)=354\).
Time = 1.10 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.20, 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.
| \(\displaystyle \int \left (d+e x+f x^2\right )^2 F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (d^2 F^{c (a+b x)}+e^2 x^2 \left (\frac {2 d f}{e^2}+1\right ) F^{c (a+b x)}+2 d e x F^{c (a+b x)}+2 e f x^3 F^{c (a+b x)}+f^2 x^4 F^{c (a+b x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {24 f^2 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {12 e f F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {24 f^2 x F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {2 \left (2 d f+e^2\right ) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {12 e f x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {12 f^2 x^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {2 x \left (2 d f+e^2\right ) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 d e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {6 e f x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {4 f^2 x^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d^2 F^{c (a+b x)}}{b c \log (F)}+\frac {x^2 \left (2 d f+e^2\right ) F^{c (a+b x)}}{b c \log (F)}+\frac {2 d e x F^{c (a+b x)}}{b c \log (F)}+\frac {2 e f x^3 F^{c (a+b x)}}{b c \log (F)}+\frac {f^2 x^4 F^{c (a+b x)}}{b c \log (F)}\) |
Input:
Int[F^(c*(a + b*x))*(d + e*x + f*x^2)^2,x]
Output:
(24*f^2*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (12*e*f*F^(c*(a + b*x)))/(b^ 4*c^4*Log[F]^4) - (24*f^2*F^(c*(a + b*x))*x)/(b^4*c^4*Log[F]^4) + (2*(e^2 + 2*d*f)*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (12*e*f*F^(c*(a + b*x))*x)/ (b^3*c^3*Log[F]^3) + (12*f^2*F^(c*(a + b*x))*x^2)/(b^3*c^3*Log[F]^3) - (2* d*e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2) - (2*(e^2 + 2*d*f)*F^(c*(a + b*x)) *x)/(b^2*c^2*Log[F]^2) - (6*e*f*F^(c*(a + b*x))*x^2)/(b^2*c^2*Log[F]^2) - (4*f^2*F^(c*(a + b*x))*x^3)/(b^2*c^2*Log[F]^2) + (d^2*F^(c*(a + b*x)))/(b* c*Log[F]) + (2*d*e*F^(c*(a + b*x))*x)/(b*c*Log[F]) + ((e^2 + 2*d*f)*F^(c*( a + b*x))*x^2)/(b*c*Log[F]) + (2*e*f*F^(c*(a + b*x))*x^3)/(b*c*Log[F]) + ( f^2*F^(c*(a + b*x))*x^4)/(b*c*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 = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.60
| method | result | size |
| gosper | \(\frac {\left (f^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+2 \ln \left (F \right )^{4} b^{4} c^{4} e f \,x^{3}+2 \ln \left (F \right )^{4} b^{4} c^{4} d f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e^{2} x^{2}+2 \ln \left (F \right )^{4} b^{4} c^{4} d e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} c^{3} e f \,x^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} d f x -2 \ln \left (F \right )^{3} b^{3} c^{3} e^{2} x -2 \ln \left (F \right )^{3} b^{3} c^{3} d e +12 \ln \left (F \right )^{2} b^{2} c^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} c^{2} e f x +4 \ln \left (F \right )^{2} b^{2} c^{2} d f +2 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}-24 \ln \left (F \right ) b c \,f^{2} x -12 \ln \left (F \right ) b c e f +24 f^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(284\) |
| risch | \(\frac {\left (f^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+2 \ln \left (F \right )^{4} b^{4} c^{4} e f \,x^{3}+2 \ln \left (F \right )^{4} b^{4} c^{4} d f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e^{2} x^{2}+2 \ln \left (F \right )^{4} b^{4} c^{4} d e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} c^{3} e f \,x^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} d f x -2 \ln \left (F \right )^{3} b^{3} c^{3} e^{2} x -2 \ln \left (F \right )^{3} b^{3} c^{3} d e +12 \ln \left (F \right )^{2} b^{2} c^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} c^{2} e f x +4 \ln \left (F \right )^{2} b^{2} c^{2} d f +2 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}-24 \ln \left (F \right ) b c \,f^{2} x -12 \ln \left (F \right ) b c e f +24 f^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(284\) |
| orering | \(\frac {\left (f^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+2 \ln \left (F \right )^{4} b^{4} c^{4} e f \,x^{3}+2 \ln \left (F \right )^{4} b^{4} c^{4} d f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e^{2} x^{2}+2 \ln \left (F \right )^{4} b^{4} c^{4} d e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} c^{3} e f \,x^{2}-4 \ln \left (F \right )^{3} b^{3} c^{3} d f x -2 \ln \left (F \right )^{3} b^{3} c^{3} e^{2} x -2 \ln \left (F \right )^{3} b^{3} c^{3} d e +12 \ln \left (F \right )^{2} b^{2} c^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} c^{2} e f x +4 \ln \left (F \right )^{2} b^{2} c^{2} d f +2 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}-24 \ln \left (F \right ) b c \,f^{2} x -12 \ln \left (F \right ) b c e f +24 f^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(284\) |
| norman | \(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-2 \ln \left (F \right )^{3} b^{3} c^{3} d e +4 \ln \left (F \right )^{2} b^{2} c^{2} d f +2 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}-12 \ln \left (F \right ) b c e f +24 f^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {\left (2 \ln \left (F \right )^{2} b^{2} c^{2} d f +\ln \left (F \right )^{2} b^{2} c^{2} e^{2}-6 \ln \left (F \right ) b c e f +12 f^{2}\right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} c^{3}}+\frac {f^{2} x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}+\frac {2 \left (\ln \left (F \right )^{3} b^{3} c^{3} d e -2 \ln \left (F \right )^{2} b^{2} c^{2} d f -\ln \left (F \right )^{2} b^{2} c^{2} e^{2}+6 \ln \left (F \right ) b c e f -12 f^{2}\right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}+\frac {2 f \left (\ln \left (F \right ) b c e -2 f \right ) x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}\) | \(302\) |
| parallelrisch | \(\frac {x^{4} F^{c \left (b x +a \right )} f^{2} \ln \left (F \right )^{4} b^{4} c^{4}+2 \ln \left (F \right )^{4} x^{3} F^{c \left (b x +a \right )} b^{4} c^{4} e f +2 \ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} d f +\ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} e^{2}+2 \ln \left (F \right )^{4} x \,F^{c \left (b x +a \right )} b^{4} c^{4} d e +\ln \left (F \right )^{4} F^{c \left (b x +a \right )} b^{4} c^{4} d^{2}-4 \ln \left (F \right )^{3} x^{3} F^{c \left (b x +a \right )} b^{3} c^{3} f^{2}-6 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} e f -4 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d f -2 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} e^{2}-2 \ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d e +12 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} f^{2}+12 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} e f +4 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d f +2 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{2}-24 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,f^{2}-12 \ln \left (F \right ) F^{c \left (b x +a \right )} b c e f +24 F^{c \left (b x +a \right )} f^{2}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(437\) |
| meijerg | \(-\frac {F^{a c} f^{2} \left (24-\frac {\left (5 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-120 b c x \ln \left (F \right )+120\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{5}\right )}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {F^{a c} f \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {F^{a c} \left (e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{4 b^{3} c^{3} \ln \left (F \right )^{3}}-\frac {F^{a c} f \left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {F^{a c} \left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{b^{3} c^{3} \ln \left (F \right )^{3}}-\frac {F^{a c} \left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{4 f \ln \left (F \right )^{2} b^{2} c^{2}}-\frac {F^{a c} \left (-e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{4 b^{3} c^{3} \ln \left (F \right )^{3}}+\frac {F^{a c} \left (-e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{4 \ln \left (F \right )^{2} b^{2} c^{2} f}-\frac {F^{a c} \left (-e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (e +\sqrt {-4 d f +e^{2}}\right )^{2} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{16 f^{2} \ln \left (F \right ) b c}\) | \(656\) |
Input:
int(F^(c*(b*x+a))*(f*x^2+e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
(f^2*x^4*ln(F)^4*b^4*c^4+2*ln(F)^4*b^4*c^4*e*f*x^3+2*ln(F)^4*b^4*c^4*d*f*x ^2+ln(F)^4*b^4*c^4*e^2*x^2+2*ln(F)^4*b^4*c^4*d*e*x+ln(F)^4*b^4*c^4*d^2-4*l n(F)^3*b^3*c^3*f^2*x^3-6*ln(F)^3*b^3*c^3*e*f*x^2-4*ln(F)^3*b^3*c^3*d*f*x-2 *ln(F)^3*b^3*c^3*e^2*x-2*ln(F)^3*b^3*c^3*d*e+12*ln(F)^2*b^2*c^2*f^2*x^2+12 *ln(F)^2*b^2*c^2*e*f*x+4*ln(F)^2*b^2*c^2*d*f+2*ln(F)^2*b^2*c^2*e^2-24*ln(F )*b*c*f^2*x-12*ln(F)*b*c*e*f+24*f^2)*F^(c*(b*x+a))/ln(F)^5/b^5/c^5
Time = 0.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.36 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {{\left ({\left (b^{4} c^{4} f^{2} x^{4} + 2 \, b^{4} c^{4} e f x^{3} + 2 \, b^{4} c^{4} d e x + b^{4} c^{4} d^{2} + {\left (b^{4} c^{4} e^{2} + 2 \, b^{4} c^{4} d f\right )} x^{2}\right )} \log \left (F\right )^{4} - 2 \, {\left (2 \, b^{3} c^{3} f^{2} x^{3} + 3 \, b^{3} c^{3} e f x^{2} + b^{3} c^{3} d e + {\left (b^{3} c^{3} e^{2} + 2 \, b^{3} c^{3} d f\right )} x\right )} \log \left (F\right )^{3} + 2 \, {\left (6 \, b^{2} c^{2} f^{2} x^{2} + 6 \, b^{2} c^{2} e f x + b^{2} c^{2} e^{2} + 2 \, b^{2} c^{2} d f\right )} \log \left (F\right )^{2} + 24 \, f^{2} - 12 \, {\left (2 \, b c f^{2} x + b c e f\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(f*x^2+e*x+d)^2,x, algorithm="fricas")
Output:
((b^4*c^4*f^2*x^4 + 2*b^4*c^4*e*f*x^3 + 2*b^4*c^4*d*e*x + b^4*c^4*d^2 + (b ^4*c^4*e^2 + 2*b^4*c^4*d*f)*x^2)*log(F)^4 - 2*(2*b^3*c^3*f^2*x^3 + 3*b^3*c ^3*e*f*x^2 + b^3*c^3*d*e + (b^3*c^3*e^2 + 2*b^3*c^3*d*f)*x)*log(F)^3 + 2*( 6*b^2*c^2*f^2*x^2 + 6*b^2*c^2*e*f*x + b^2*c^2*e^2 + 2*b^2*c^2*d*f)*log(F)^ 2 + 24*f^2 - 12*(2*b*c*f^2*x + b*c*e*f)*log(F))*F^(b*c*x + a*c)/(b^5*c^5*l og(F)^5)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (180) = 360\).
Time = 0.12 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.17 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{2} \log {\left (F \right )}^{4} + 2 b^{4} c^{4} d e x \log {\left (F \right )}^{4} + 2 b^{4} c^{4} d f x^{2} \log {\left (F \right )}^{4} + b^{4} c^{4} e^{2} x^{2} \log {\left (F \right )}^{4} + 2 b^{4} c^{4} e f x^{3} \log {\left (F \right )}^{4} + b^{4} c^{4} f^{2} x^{4} \log {\left (F \right )}^{4} - 2 b^{3} c^{3} d e \log {\left (F \right )}^{3} - 4 b^{3} c^{3} d f x \log {\left (F \right )}^{3} - 2 b^{3} c^{3} e^{2} x \log {\left (F \right )}^{3} - 6 b^{3} c^{3} e f x^{2} \log {\left (F \right )}^{3} - 4 b^{3} c^{3} f^{2} x^{3} \log {\left (F \right )}^{3} + 4 b^{2} c^{2} d f \log {\left (F \right )}^{2} + 2 b^{2} c^{2} e^{2} \log {\left (F \right )}^{2} + 12 b^{2} c^{2} e f x \log {\left (F \right )}^{2} + 12 b^{2} c^{2} f^{2} x^{2} \log {\left (F \right )}^{2} - 12 b c e f \log {\left (F \right )} - 24 b c f^{2} x \log {\left (F \right )} + 24 f^{2}\right )}{b^{5} c^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} c^{5} \log {\left (F \right )}^{5} \neq 0 \\d^{2} x + d e x^{2} + \frac {e f x^{4}}{2} + \frac {f^{2} x^{5}}{5} + x^{3} \cdot \left (\frac {2 d f}{3} + \frac {e^{2}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(F**((b*x+a)*c)*(f*x**2+e*x+d)**2,x)
Output:
Piecewise((F**(c*(a + b*x))*(b**4*c**4*d**2*log(F)**4 + 2*b**4*c**4*d*e*x* log(F)**4 + 2*b**4*c**4*d*f*x**2*log(F)**4 + b**4*c**4*e**2*x**2*log(F)**4 + 2*b**4*c**4*e*f*x**3*log(F)**4 + b**4*c**4*f**2*x**4*log(F)**4 - 2*b**3 *c**3*d*e*log(F)**3 - 4*b**3*c**3*d*f*x*log(F)**3 - 2*b**3*c**3*e**2*x*log (F)**3 - 6*b**3*c**3*e*f*x**2*log(F)**3 - 4*b**3*c**3*f**2*x**3*log(F)**3 + 4*b**2*c**2*d*f*log(F)**2 + 2*b**2*c**2*e**2*log(F)**2 + 12*b**2*c**2*e* f*x*log(F)**2 + 12*b**2*c**2*f**2*x**2*log(F)**2 - 12*b*c*e*f*log(F) - 24* b*c*f**2*x*log(F) + 24*f**2)/(b**5*c**5*log(F)**5), Ne(b**5*c**5*log(F)**5 , 0)), (d**2*x + d*e*x**2 + e*f*x**4/2 + f**2*x**5/5 + x**3*(2*d*f/3 + e** 2/3), True))
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (177) = 354\).
Time = 0.04 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.03 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {F^{b c x + a c} d^{2}}{b c \log \left (F\right )} + \frac {2 \, {\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d f}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} e f}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} f^{2}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(f*x^2+e*x+d)^2,x, algorithm="maxima")
Output:
F^(b*c*x + a*c)*d^2/(b*c*log(F)) + 2*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b *c*x)*d*e/(b^2*c^2*log(F)^2) + (F^(a*c)*b^2*c^2*x^2*log(F)^2 - 2*F^(a*c)*b *c*x*log(F) + 2*F^(a*c))*F^(b*c*x)*e^2/(b^3*c^3*log(F)^3) + 2*(F^(a*c)*b^2 *c^2*x^2*log(F)^2 - 2*F^(a*c)*b*c*x*log(F) + 2*F^(a*c))*F^(b*c*x)*d*f/(b^3 *c^3*log(F)^3) + 2*(F^(a*c)*b^3*c^3*x^3*log(F)^3 - 3*F^(a*c)*b^2*c^2*x^2*l og(F)^2 + 6*F^(a*c)*b*c*x*log(F) - 6*F^(a*c))*F^(b*c*x)*e*f/(b^4*c^4*log(F )^4) + (F^(a*c)*b^4*c^4*x^4*log(F)^4 - 4*F^(a*c)*b^3*c^3*x^3*log(F)^3 + 12 *F^(a*c)*b^2*c^2*x^2*log(F)^2 - 24*F^(a*c)*b*c*x*log(F) + 24*F^(a*c))*F^(b *c*x)*f^2/(b^5*c^5*log(F)^5)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 9704, normalized size of antiderivative = 54.82 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(F^((b*x+a)*c)*(f*x^2+e*x+d)^2,x, algorithm="giac")
Output:
-((2*(2*pi^3*b^4*c^4*f^2*x^4*log(abs(F))*sgn(F) - 2*pi*b^4*c^4*f^2*x^4*log (abs(F))^3*sgn(F) - 2*pi^3*b^4*c^4*f^2*x^4*log(abs(F)) + 2*pi*b^4*c^4*f^2* x^4*log(abs(F))^3 + 4*pi^3*b^4*c^4*e*f*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*c ^4*e*f*x^3*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*e*f*x^3*log(abs(F)) + 4*p i*b^4*c^4*e*f*x^3*log(abs(F))^3 + 2*pi^3*b^4*c^4*e^2*x^2*log(abs(F))*sgn(F ) + 4*pi^3*b^4*c^4*d*f*x^2*log(abs(F))*sgn(F) - 2*pi*b^4*c^4*e^2*x^2*log(a bs(F))^3*sgn(F) - 4*pi*b^4*c^4*d*f*x^2*log(abs(F))^3*sgn(F) - 2*pi^3*b^4*c ^4*e^2*x^2*log(abs(F)) - 4*pi^3*b^4*c^4*d*f*x^2*log(abs(F)) + 2*pi*b^4*c^4 *e^2*x^2*log(abs(F))^3 + 4*pi*b^4*c^4*d*f*x^2*log(abs(F))^3 + 4*pi^3*b^4*c ^4*d*e*x*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*d*e*x*log(abs(F))^3*sgn(F) - 4* pi^3*b^4*c^4*d*e*x*log(abs(F)) + 4*pi*b^4*c^4*d*e*x*log(abs(F))^3 - 2*pi^3 *b^3*c^3*f^2*x^3*sgn(F) + 2*pi^3*b^4*c^4*d^2*log(abs(F))*sgn(F) + 6*pi*b^3 *c^3*f^2*x^3*log(abs(F))^2*sgn(F) - 2*pi*b^4*c^4*d^2*log(abs(F))^3*sgn(F) + 2*pi^3*b^3*c^3*f^2*x^3 - 2*pi^3*b^4*c^4*d^2*log(abs(F)) - 6*pi*b^3*c^3*f ^2*x^3*log(abs(F))^2 + 2*pi*b^4*c^4*d^2*log(abs(F))^3 - 3*pi^3*b^3*c^3*e*f *x^2*sgn(F) + 9*pi*b^3*c^3*e*f*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*e *f*x^2 - 9*pi*b^3*c^3*e*f*x^2*log(abs(F))^2 - pi^3*b^3*c^3*e^2*x*sgn(F) - 2*pi^3*b^3*c^3*d*f*x*sgn(F) + 3*pi*b^3*c^3*e^2*x*log(abs(F))^2*sgn(F) + 6* pi*b^3*c^3*d*f*x*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*e^2*x + 2*pi^3*b^3*c^ 3*d*f*x - 3*pi*b^3*c^3*e^2*x*log(abs(F))^2 - 6*pi*b^3*c^3*d*f*x*log(abs...
Time = 23.14 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.60 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^4\,c^4\,d^2\,{\ln \left (F\right )}^4+2\,b^4\,c^4\,d\,e\,x\,{\ln \left (F\right )}^4+2\,b^4\,c^4\,d\,f\,x^2\,{\ln \left (F\right )}^4+b^4\,c^4\,e^2\,x^2\,{\ln \left (F\right )}^4+2\,b^4\,c^4\,e\,f\,x^3\,{\ln \left (F\right )}^4+b^4\,c^4\,f^2\,x^4\,{\ln \left (F\right )}^4-2\,b^3\,c^3\,d\,e\,{\ln \left (F\right )}^3-4\,b^3\,c^3\,d\,f\,x\,{\ln \left (F\right )}^3-2\,b^3\,c^3\,e^2\,x\,{\ln \left (F\right )}^3-6\,b^3\,c^3\,e\,f\,x^2\,{\ln \left (F\right )}^3-4\,b^3\,c^3\,f^2\,x^3\,{\ln \left (F\right )}^3+4\,b^2\,c^2\,d\,f\,{\ln \left (F\right )}^2+2\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+12\,b^2\,c^2\,e\,f\,x\,{\ln \left (F\right )}^2+12\,b^2\,c^2\,f^2\,x^2\,{\ln \left (F\right )}^2-12\,b\,c\,e\,f\,\ln \left (F\right )-24\,b\,c\,f^2\,x\,\ln \left (F\right )+24\,f^2\right )}{b^5\,c^5\,{\ln \left (F\right )}^5} \] Input:
int(F^(c*(a + b*x))*(d + e*x + f*x^2)^2,x)
Output:
(F^(a*c + b*c*x)*(24*f^2 + b^4*c^4*d^2*log(F)^4 + 2*b^2*c^2*e^2*log(F)^2 - 24*b*c*f^2*x*log(F) - 2*b^3*c^3*e^2*x*log(F)^3 + b^4*c^4*e^2*x^2*log(F)^4 + 12*b^2*c^2*f^2*x^2*log(F)^2 - 4*b^3*c^3*f^2*x^3*log(F)^3 + b^4*c^4*f^2* x^4*log(F)^4 - 12*b*c*e*f*log(F) - 2*b^3*c^3*d*e*log(F)^3 + 4*b^2*c^2*d*f* log(F)^2 + 2*b^4*c^4*d*e*x*log(F)^4 - 4*b^3*c^3*d*f*x*log(F)^3 + 12*b^2*c^ 2*e*f*x*log(F)^2 + 2*b^4*c^4*d*f*x^2*log(F)^4 - 6*b^3*c^3*e*f*x^2*log(F)^3 + 2*b^4*c^4*e*f*x^3*log(F)^4))/(b^5*c^5*log(F)^5)
Time = 0.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.60 \[ \int F^{c (a+b x)} \left (d+e x+f x^2\right )^2 \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2}+2 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d e x +2 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d f \,x^{2}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} e^{2} x^{2}+2 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} e f \,x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} f^{2} x^{4}-2 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e -4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d f x -2 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} x -6 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e f \,x^{2}-4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} f^{2} x^{3}+4 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d f +2 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2}+12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e f x +12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} f^{2} x^{2}-12 \,\mathrm {log}\left (f \right ) b c e f -24 \,\mathrm {log}\left (f \right ) b c \,f^{2} x +24 f^{2}\right )}{\mathrm {log}\left (f \right )^{5} b^{5} c^{5}} \] Input:
int(F^((b*x+a)*c)*(f*x^2+e*x+d)^2,x)
Output:
(f**(a*c + b*c*x)*(log(f)**4*b**4*c**4*d**2 + 2*log(f)**4*b**4*c**4*d*e*x + 2*log(f)**4*b**4*c**4*d*f*x**2 + log(f)**4*b**4*c**4*e**2*x**2 + 2*log(f )**4*b**4*c**4*e*f*x**3 + log(f)**4*b**4*c**4*f**2*x**4 - 2*log(f)**3*b**3 *c**3*d*e - 4*log(f)**3*b**3*c**3*d*f*x - 2*log(f)**3*b**3*c**3*e**2*x - 6 *log(f)**3*b**3*c**3*e*f*x**2 - 4*log(f)**3*b**3*c**3*f**2*x**3 + 4*log(f) **2*b**2*c**2*d*f + 2*log(f)**2*b**2*c**2*e**2 + 12*log(f)**2*b**2*c**2*e* f*x + 12*log(f)**2*b**2*c**2*f**2*x**2 - 12*log(f)*b*c*e*f - 24*log(f)*b*c *f**2*x + 24*f**2))/(log(f)**5*b**5*c**5)