\(\int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx\) [66]

Optimal result

Integrand size = 22, antiderivative size = 328 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)} \]

[Out]

24*f^2*F^(b*d*x+b*c+a)/b^5/d^5/ln(F)^5-12*e*f*F^(b*d*x+b*c+a)/b^4/d^4/ln(F)^4-24*f^2*F^(b*d*x+b*c+a)*x/b^4/d^4
/ln(F)^4+2*e^2*F^(b*d*x+b*c+a)/b^3/d^3/ln(F)^3+12*e*f*F^(b*d*x+b*c+a)*x/b^3/d^3/ln(F)^3+12*f^2*F^(b*d*x+b*c+a)
*x^2/b^3/d^3/ln(F)^3-2*e^2*F^(b*d*x+b*c+a)*x/b^2/d^2/ln(F)^2-6*e*f*F^(b*d*x+b*c+a)*x^2/b^2/d^2/ln(F)^2-4*f^2*F
^(b*d*x+b*c+a)*x^3/b^2/d^2/ln(F)^2+e^2*F^(b*d*x+b*c+a)*x^2/b/d/ln(F)+2*e*f*F^(b*d*x+b*c+a)*x^3/b/d/ln(F)+f^2*F
^(b*d*x+b*c+a)*x^4/b/d/ln(F)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2227, 2207, 2225} \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^4 F^{a+b c+b d x}}{b d \log (F)} \]

[In]

Int[F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x]

[Out]

(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*Log[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*Log[F])

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^2 F^{a+b c+b d x} x^2+2 e f F^{a+b c+b d x} x^3+f^2 F^{a+b c+b d x} x^4\right ) \, dx \\ & = e^2 \int F^{a+b c+b d x} x^2 \, dx+(2 e f) \int F^{a+b c+b d x} x^3 \, dx+f^2 \int F^{a+b c+b d x} x^4 \, dx \\ & = \frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}-\frac {\left (2 e^2\right ) \int F^{a+b c+b d x} x \, dx}{b d \log (F)}-\frac {(6 e f) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)}-\frac {\left (4 f^2\right ) \int F^{a+b c+b d x} x^3 \, dx}{b d \log (F)} \\ & = -\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {\left (2 e^2\right ) \int F^{a+b c+b d x} \, dx}{b^2 d^2 \log ^2(F)}+\frac {(12 e f) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)}+\frac {\left (12 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b^2 d^2 \log ^2(F)} \\ & = \frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}-\frac {(12 e f) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)}-\frac {\left (24 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^3 d^3 \log ^3(F)} \\ & = -\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 F^{a+b c+b d x} 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 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {\left (24 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^4 d^4 \log ^4(F)} \\ & = \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 F^{a+b c+b d x} 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 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.37 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {F^{a+b (c+d x)} \left (24 f^2-12 b d f (e+2 f x) \log (F)+2 b^2 d^2 \left (e^2+6 e f x+6 f^2 x^2\right ) \log ^2(F)-2 b^3 d^3 x \left (e^2+3 e f x+2 f^2 x^2\right ) \log ^3(F)+b^4 d^4 x^2 (e+f x)^2 \log ^4(F)\right )}{b^5 d^5 \log ^5(F)} \]

[In]

Integrate[F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x]

[Out]

(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)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {\left (\ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}+2 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}+\ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}-2 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x +12 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} d^{2} e f x +2 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) b d \,f^{2} x -12 e f \ln \left (F \right ) b d +24 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{5} b^{5} d^{5}}\)	\(197\)
risch	\(\frac {\left (\ln \left (F \right )^{4} b^{4} d^{4} f^{2} x^{4}+2 \ln \left (F \right )^{4} b^{4} d^{4} e f \,x^{3}+\ln \left (F \right )^{4} b^{4} d^{4} e^{2} x^{2}-4 \ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}-6 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}-2 \ln \left (F \right )^{3} b^{3} d^{3} e^{2} x +12 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}+12 \ln \left (F \right )^{2} b^{2} d^{2} e f x +2 \ln \left (F \right )^{2} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) b d \,f^{2} x -12 e f \ln \left (F \right ) b d +24 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{5} b^{5} d^{5}}\)	\(197\)
meijerg	\(-\frac {F^{c b +a} f^{2} \left (24-\frac {\left (5 b^{4} d^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-120 b d x \ln \left (F \right )+120\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{5}\right )}{\ln \left (F \right )^{5} b^{5} d^{5}}+\frac {2 F^{c b +a} f e \left (6-\frac {\left (-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 b d x \ln \left (F \right )+24\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{4}\right )}{b^{4} d^{4} \ln \left (F \right )^{4}}-\frac {F^{c b +a} e^{2} \left (2-\frac {\left (3 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-6 b d x \ln \left (F \right )+6\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{3}\right )}{b^{3} d^{3} \ln \left (F \right )^{3}}\)	\(217\)
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	\(\frac {\ln \left (F \right )^{4} x^{4} F^{b d x +c b +a} b^{4} d^{4} f^{2}+2 \ln \left (F \right )^{4} x^{3} F^{b d x +c b +a} b^{4} d^{4} e f +\ln \left (F \right )^{4} x^{2} F^{b d x +c b +a} b^{4} d^{4} e^{2}-4 \ln \left (F \right )^{3} x^{3} F^{b d x +c b +a} b^{3} d^{3} f^{2}-6 \ln \left (F \right )^{3} x^{2} F^{b d x +c b +a} b^{3} d^{3} e f -2 \ln \left (F \right )^{3} x \,F^{b d x +c b +a} b^{3} d^{3} e^{2}+12 \ln \left (F \right )^{2} x^{2} F^{b d x +c b +a} b^{2} d^{2} f^{2}+12 \ln \left (F \right )^{2} x \,F^{b d x +c b +a} b^{2} d^{2} e f +2 \ln \left (F \right )^{2} F^{b d x +c b +a} b^{2} d^{2} e^{2}-24 \ln \left (F \right ) x \,F^{b d x +c b +a} b d \,f^{2}-12 \ln \left (F \right ) F^{b d x +c b +a} b d e f +24 F^{b d x +c b +a} f^{2}}{\ln \left (F \right )^{5} b^{5} d^{5}}\)	\(318\)

[In]

int(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x,method=_RETURNVERBOSE)

[Out]

(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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.54 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {{\left ({\left (b^{4} d^{4} f^{2} x^{4} + 2 \, b^{4} d^{4} e f x^{3} + b^{4} d^{4} e^{2} x^{2}\right )} \log \left (F\right )^{4} - 2 \, {\left (2 \, b^{3} d^{3} f^{2} x^{3} + 3 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} + 2 \, {\left (6 \, b^{2} d^{2} f^{2} x^{2} + 6 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 24 \, f^{2} - 12 \, {\left (2 \, b d f^{2} x + b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{5} d^{5} \log \left (F\right )^{5}} \]

[In]

integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="fricas")

[Out]

((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)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.79 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{4} d^{4} e^{2} x^{2} \log {\left (F \right )}^{4} + 2 b^{4} d^{4} e f x^{3} \log {\left (F \right )}^{4} + b^{4} d^{4} f^{2} x^{4} \log {\left (F \right )}^{4} - 2 b^{3} d^{3} e^{2} x \log {\left (F \right )}^{3} - 6 b^{3} d^{3} e f x^{2} \log {\left (F \right )}^{3} - 4 b^{3} d^{3} f^{2} x^{3} \log {\left (F \right )}^{3} + 2 b^{2} d^{2} e^{2} \log {\left (F \right )}^{2} + 12 b^{2} d^{2} e f x \log {\left (F \right )}^{2} + 12 b^{2} d^{2} f^{2} x^{2} \log {\left (F \right )}^{2} - 12 b d e f \log {\left (F \right )} - 24 b d f^{2} x \log {\left (F \right )} + 24 f^{2}\right )}{b^{5} d^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d^{5} \log {\left (F \right )}^{5} \neq 0 \\\frac {e^{2} x^{3}}{3} + \frac {e f x^{4}}{2} + \frac {f^{2} x^{5}}{5} & \text {otherwise} \end {cases} \]

[In]

integrate(F**(a+b*(d*x+c))*x**2*(f*x+e)**2,x)

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.80 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} e f}{b^{4} d^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{b c + a} b d x \log \left (F\right ) + 24 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{5} d^{5} \log \left (F\right )^{5}} \]

[In]

integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="maxima")

[Out]

(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*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) + 24*F^(b*c + a))*F^(b*d*x)*f^2/
(b^5*d^5*log(F)^5)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 6582, normalized size of antiderivative = 20.07 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\text {Too large to display} \]

[In]

integrate(F^(a+b*(d*x+c))*x^2*(f*x+e)^2,x, algorithm="giac")

[Out]

-((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*pi*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(a
bs(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*lo
g(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*log(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*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^
4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F))
+ 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2) - (pi^4*b^4*d^4*f^2*x^4*sgn(F) - 6*pi^2*b^4*d^4*
f^2*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*f^2*x^4 + 6*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 - 2*b^4*d^4*f^2*x^4
*log(abs(F))^4 + 2*pi^4*b^4*d^4*e*f*x^3*sgn(F) - 12*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2*sgn(F) - 2*pi^4*b^4*d^4
*e*f*x^3 + 12*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2 - 4*b^4*d^4*e*f*x^3*log(abs(F))^4 + pi^4*b^4*d^4*e^2*x^2*sgn(
F) - 6*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*e^2*x^2 + 6*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2
- 2*b^4*d^4*e^2*x^2*log(abs(F))^4 + 12*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) - 12*pi^2*b^3*d^3*f^2*x^3*log(
abs(F)) + 8*b^3*d^3*f^2*x^3*log(abs(F))^3 + 18*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 18*pi^2*b^3*d^3*e*f*x
^2*log(abs(F)) + 12*b^3*d^3*e*f*x^2*log(abs(F))^3 + 6*pi^2*b^3*d^3*e^2*x*log(abs(F))*sgn(F) - 6*pi^2*b^3*d^3*e
^2*x*log(abs(F)) + 4*b^3*d^3*e^2*x*log(abs(F))^3 - 12*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 12*pi^2*b^2*d^2*f^2*x^2 -
24*b^2*d^2*f^2*x^2*log(abs(F))^2 - 12*pi^2*b^2*d^2*e*f*x*sgn(F) + 12*pi^2*b^2*d^2*e*f*x - 24*b^2*d^2*e*f*x*log
(abs(F))^2 - 2*pi^2*b^2*d^2*e^2*sgn(F) + 2*pi^2*b^2*d^2*e^2 - 4*b^2*d^2*e^2*log(abs(F))^2 + 48*b*d*f^2*x*log(a
bs(F)) + 24*b*d*e*f*log(abs(F)) - 48*f^2)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*s
gn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)/((pi^5*b^5*d^5*s
gn(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)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(a
bs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2))*co
s(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a) - ((pi^4*
b^4*d^4*f^2*x^4*sgn(F) - 6*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*f^2*x^4 + 6*pi^2*b^4*d^4*f
^2*x^4*log(abs(F))^2 - 2*b^4*d^4*f^2*x^4*log(abs(F))^4 + 2*pi^4*b^4*d^4*e*f*x^3*sgn(F) - 12*pi^2*b^4*d^4*e*f*x
^3*log(abs(F))^2*sgn(F) - 2*pi^4*b^4*d^4*e*f*x^3 + 12*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2 - 4*b^4*d^4*e*f*x^3*l
og(abs(F))^4 + pi^4*b^4*d^4*e^2*x^2*sgn(F) - 6*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*e^2*x^
2 + 6*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2 - 2*b^4*d^4*e^2*x^2*log(abs(F))^4 + 12*pi^2*b^3*d^3*f^2*x^3*log(abs(F
))*sgn(F) - 12*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 8*b^3*d^3*f^2*x^3*log(abs(F))^3 + 18*pi^2*b^3*d^3*e*f*x^2*lo
g(abs(F))*sgn(F) - 18*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) + 12*b^3*d^3*e*f*x^2*log(abs(F))^3 + 6*pi^2*b^3*d^3*e^2
*x*log(abs(F))*sgn(F) - 6*pi^2*b^3*d^3*e^2*x*log(abs(F)) + 4*b^3*d^3*e^2*x*log(abs(F))^3 - 12*pi^2*b^2*d^2*f^2
*x^2*sgn(F) + 12*pi^2*b^2*d^2*f^2*x^2 - 24*b^2*d^2*f^2*x^2*log(abs(F))^2 - 12*pi^2*b^2*d^2*e*f*x*sgn(F) + 12*p
i^2*b^2*d^2*e*f*x - 24*b^2*d^2*e*f*x*log(abs(F))^2 - 2*pi^2*b^2*d^2*e^2*sgn(F) + 2*pi^2*b^2*d^2*e^2 - 4*b^2*d^
2*e^2*log(abs(F))^2 + 48*b*d*f^2*x*log(abs(F)) + 24*b*d*e*f*log(abs(F)) - 48*f^2)*(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*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4
*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*
d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2) + 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*pi*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*log(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*p
i*b*d*e*f*sgn(F) - 6*pi*b*d*e*f)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5
*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)/((pi^5*b^5*d^5*sgn(F) - 1
0*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(a
bs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*
sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2))*sin(-1/2*pi
*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a))*e^(b*d*x*log(abs(
F)) + b*c*log(abs(F)) + a*log(abs(F))) - 8*I*((I*pi^4*b^4*d^4*f^2*x^4*sgn(F) - 4*pi^3*b^4*d^4*f^2*x^4*log(abs(
F))*sgn(F) - 6*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) + 4*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - I*pi^
4*b^4*d^4*f^2*x^4 + 4*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 6*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 - 4*pi*b^4*d^4
*f^2*x^4*log(abs(F))^3 - 2*I*b^4*d^4*f^2*x^4*log(abs(F))^4 + 2*I*pi^4*b^4*d^4*e*f*x^3*sgn(F) - 8*pi^3*b^4*d^4*
e*f*x^3*log(abs(F))*sgn(F) - 12*I*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2*sgn(F) + 8*pi*b^4*d^4*e*f*x^3*log(abs(F))
^3*sgn(F) - 2*I*pi^4*b^4*d^4*e*f*x^3 + 8*pi^3*b^4*d^4*e*f*x^3*log(abs(F)) + 12*I*pi^2*b^4*d^4*e*f*x^3*log(abs(
F))^2 - 8*pi*b^4*d^4*e*f*x^3*log(abs(F))^3 - 4*I*b^4*d^4*e*f*x^3*log(abs(F))^4 + I*pi^4*b^4*d^4*e^2*x^2*sgn(F)
- 4*pi^3*b^4*d^4*e^2*x^2*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2*sgn(F) + 4*pi*b^4*d^4*e^
2*x^2*log(abs(F))^3*sgn(F) - I*pi^4*b^4*d^4*e^2*x^2 + 4*pi^3*b^4*d^4*e^2*x^2*log(abs(F)) + 6*I*pi^2*b^4*d^4*e^
2*x^2*log(abs(F))^2 - 4*pi*b^4*d^4*e^2*x^2*log(abs(F))^3 - 2*I*b^4*d^4*e^2*x^2*log(abs(F))^4 + 4*pi^3*b^3*d^3*
f^2*x^3*sgn(F) + 12*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) - 12*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) - 4
*pi^3*b^3*d^3*f^2*x^3 - 12*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 12*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 8*I*b^3*
d^3*f^2*x^3*log(abs(F))^3 + 6*pi^3*b^3*d^3*e*f*x^2*sgn(F) + 18*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 18*
pi*b^3*d^3*e*f*x^2*log(abs(F))^2*sgn(F) - 6*pi^3*b^3*d^3*e*f*x^2 - 18*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) + 18*
pi*b^3*d^3*e*f*x^2*log(abs(F))^2 + 12*I*b^3*d^3*e*f*x^2*log(abs(F))^3 + 2*pi^3*b^3*d^3*e^2*x*sgn(F) + 6*I*pi^2
*b^3*d^3*e^2*x*log(abs(F))*sgn(F) - 6*pi*b^3*d^3*e^2*x*log(abs(F))^2*sgn(F) - 2*pi^3*b^3*d^3*e^2*x - 6*I*pi^2*
b^3*d^3*e^2*x*log(abs(F)) + 6*pi*b^3*d^3*e^2*x*log(abs(F))^2 + 4*I*b^3*d^3*e^2*x*log(abs(F))^3 - 12*I*pi^2*b^2
*d^2*f^2*x^2*sgn(F) + 24*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2*d^2*f^2*x^2 - 24*pi*b^2*d^2*f^2
*x^2*log(abs(F)) - 24*I*b^2*d^2*f^2*x^2*log(abs(F))^2 - 12*I*pi^2*b^2*d^2*e*f*x*sgn(F) + 24*pi*b^2*d^2*e*f*x*l
og(abs(F))*sgn(F) + 12*I*pi^2*b^2*d^2*e*f*x - 24*pi*b^2*d^2*e*f*x*log(abs(F)) - 24*I*b^2*d^2*e*f*x*log(abs(F))
^2 - 2*I*pi^2*b^2*d^2*e^2*sgn(F) + 4*pi*b^2*d^2*e^2*log(abs(F))*sgn(F) + 2*I*pi^2*b^2*d^2*e^2 - 4*pi*b^2*d^2*e
^2*log(abs(F)) - 4*I*b^2*d^2*e^2*log(abs(F))^2 - 24*pi*b*d*f^2*x*sgn(F) + 24*pi*b*d*f^2*x + 48*I*b*d*f^2*x*log
(abs(F)) - 12*pi*b*d*e*f*sgn(F) + 12*pi*b*d*e*f + 24*I*b*d*e*f*log(abs(F)) - 48*I*f^2)*e^(1/2*I*pi*b*d*x*sgn(F
) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F) - 1/2*I*pi*a)/(16*I*pi^5*b^5*d^5*s
gn(F) - 80*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 160*I*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 160*pi^2*b^5*d^5*log(ab
s(F))^3*sgn(F) + 80*I*pi*b^5*d^5*log(abs(F))^4*sgn(F) - 16*I*pi^5*b^5*d^5 + 80*pi^4*b^5*d^5*log(abs(F)) + 160*
I*pi^3*b^5*d^5*log(abs(F))^2 - 160*pi^2*b^5*d^5*log(abs(F))^3 - 80*I*pi*b^5*d^5*log(abs(F))^4 + 32*b^5*d^5*log
(abs(F))^5) - (I*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 4*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*d^4*f^
2*x^4*log(abs(F))^2*sgn(F) - 4*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - I*pi^4*b^4*d^4*f^2*x^4 - 4*pi^3*b^4*d
^4*f^2*x^4*log(abs(F)) + 6*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 4*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 - 2*I*b^4
*d^4*f^2*x^4*log(abs(F))^4 + 2*I*pi^4*b^4*d^4*e*f*x^3*sgn(F) + 8*pi^3*b^4*d^4*e*f*x^3*log(abs(F))*sgn(F) - 12*
I*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2*sgn(F) - 8*pi*b^4*d^4*e*f*x^3*log(abs(F))^3*sgn(F) - 2*I*pi^4*b^4*d^4*e*f
*x^3 - 8*pi^3*b^4*d^4*e*f*x^3*log(abs(F)) + 12*I*pi^2*b^4*d^4*e*f*x^3*log(abs(F))^2 + 8*pi*b^4*d^4*e*f*x^3*log
(abs(F))^3 - 4*I*b^4*d^4*e*f*x^3*log(abs(F))^4 + I*pi^4*b^4*d^4*e^2*x^2*sgn(F) + 4*pi^3*b^4*d^4*e^2*x^2*log(ab
s(F))*sgn(F) - 6*I*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2*sgn(F) - 4*pi*b^4*d^4*e^2*x^2*log(abs(F))^3*sgn(F) - I*p
i^4*b^4*d^4*e^2*x^2 - 4*pi^3*b^4*d^4*e^2*x^2*log(abs(F)) + 6*I*pi^2*b^4*d^4*e^2*x^2*log(abs(F))^2 + 4*pi*b^4*d
^4*e^2*x^2*log(abs(F))^3 - 2*I*b^4*d^4*e^2*x^2*log(abs(F))^4 - 4*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 12*I*pi^2*b^3*d
^3*f^2*x^3*log(abs(F))*sgn(F) + 12*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 4*pi^3*b^3*d^3*f^2*x^3 - 12*I*pi^
2*b^3*d^3*f^2*x^3*log(abs(F)) - 12*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 8*I*b^3*d^3*f^2*x^3*log(abs(F))^3 - 6*pi
^3*b^3*d^3*e*f*x^2*sgn(F) + 18*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) + 18*pi*b^3*d^3*e*f*x^2*log(abs(F))^2
*sgn(F) + 6*pi^3*b^3*d^3*e*f*x^2 - 18*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) - 18*pi*b^3*d^3*e*f*x^2*log(abs(F))^2
+ 12*I*b^3*d^3*e*f*x^2*log(abs(F))^3 - 2*pi^3*b^3*d^3*e^2*x*sgn(F) + 6*I*pi^2*b^3*d^3*e^2*x*log(abs(F))*sgn(F
) + 6*pi*b^3*d^3*e^2*x*log(abs(F))^2*sgn(F) + 2*pi^3*b^3*d^3*e^2*x - 6*I*pi^2*b^3*d^3*e^2*x*log(abs(F)) - 6*pi
*b^3*d^3*e^2*x*log(abs(F))^2 + 4*I*b^3*d^3*e^2*x*log(abs(F))^3 - 12*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 24*pi*b^2*
d^2*f^2*x^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2*d^2*f^2*x^2 + 24*pi*b^2*d^2*f^2*x^2*log(abs(F)) - 24*I*b^2*d^2*
f^2*x^2*log(abs(F))^2 - 12*I*pi^2*b^2*d^2*e*f*x*sgn(F) - 24*pi*b^2*d^2*e*f*x*log(abs(F))*sgn(F) + 12*I*pi^2*b^
2*d^2*e*f*x + 24*pi*b^2*d^2*e*f*x*log(abs(F)) - 24*I*b^2*d^2*e*f*x*log(abs(F))^2 - 2*I*pi^2*b^2*d^2*e^2*sgn(F)
- 4*pi*b^2*d^2*e^2*log(abs(F))*sgn(F) + 2*I*pi^2*b^2*d^2*e^2 + 4*pi*b^2*d^2*e^2*log(abs(F)) - 4*I*b^2*d^2*e^2
*log(abs(F))^2 + 24*pi*b*d*f^2*x*sgn(F) - 24*pi*b*d*f^2*x + 48*I*b*d*f^2*x*log(abs(F)) + 12*pi*b*d*e*f*sgn(F)
- 12*pi*b*d*e*f + 24*I*b*d*e*f*log(abs(F)) - 48*I*f^2)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2*I*pi*b*d*x - 1/2*I*pi*b
*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(-16*I*pi^5*b^5*d^5*sgn(F) - 80*pi^4*b^5*d^5*log(ab
s(F))*sgn(F) + 160*I*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 160*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 80*I*pi*b^5*d
^5*log(abs(F))^4*sgn(F) + 16*I*pi^5*b^5*d^5 + 80*pi^4*b^5*d^5*log(abs(F)) - 160*I*pi^3*b^5*d^5*log(abs(F))^2 -
160*pi^2*b^5*d^5*log(abs(F))^3 + 80*I*pi*b^5*d^5*log(abs(F))^4 + 32*b^5*d^5*log(abs(F))^5))*e^(b*d*x*log(abs(
F)) + b*c*log(abs(F)) + a*log(abs(F)))

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.60 \[ \int F^{a+b (c+d x)} x^2 (e+f x)^2 \, dx=\frac {F^{a+b\,c+b\,d\,x}\,\left (b^4\,d^4\,e^2\,x^2\,{\ln \left (F\right )}^4+2\,b^4\,d^4\,e\,f\,x^3\,{\ln \left (F\right )}^4+b^4\,d^4\,f^2\,x^4\,{\ln \left (F\right )}^4-2\,b^3\,d^3\,e^2\,x\,{\ln \left (F\right )}^3-6\,b^3\,d^3\,e\,f\,x^2\,{\ln \left (F\right )}^3-4\,b^3\,d^3\,f^2\,x^3\,{\ln \left (F\right )}^3+2\,b^2\,d^2\,e^2\,{\ln \left (F\right )}^2+12\,b^2\,d^2\,e\,f\,x\,{\ln \left (F\right )}^2+12\,b^2\,d^2\,f^2\,x^2\,{\ln \left (F\right )}^2-12\,b\,d\,e\,f\,\ln \left (F\right )-24\,b\,d\,f^2\,x\,\ln \left (F\right )+24\,f^2\right )}{b^5\,d^5\,{\ln \left (F\right )}^5} \]

[In]

int(F^(a + b*(c + d*x))*x^2*(e + f*x)^2,x)

[Out]

(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)