\(\int F^{a+b (c+d x)} x (e+f x)^2 \, dx\) [67]

Optimal result

Integrand size = 20, antiderivative size = 242 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=-\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2227, 2207, 2225} \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=-\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

[In]

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

[Out]

(-6*f^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) + (4*e*f*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (6*f^2*F^(a
+ b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) - (e^2*F^(a + b*c + b*d*x))/(b^2*d^2*Log[F]^2) - (4*e*f*F^(a + b*c + b*d
*x)*x)/(b^2*d^2*Log[F]^2) - (3*f^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x)/(
b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^2)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^3)/(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 e f F^{a+b c+b d x} x^2+f^2 F^{a+b c+b d x} x^3\right ) \, dx \\ & = e^2 \int F^{a+b c+b d x} x \, dx+(2 e f) \int F^{a+b c+b d x} x^2 \, dx+f^2 \int F^{a+b c+b d x} x^3 \, dx \\ & = \frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}-\frac {e^2 \int F^{a+b c+b d x} \, dx}{b d \log (F)}-\frac {(4 e f) \int F^{a+b c+b d x} x \, dx}{b d \log (F)}-\frac {\left (3 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)} \\ & = -\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {(4 e f) \int F^{a+b c+b d x} \, dx}{b^2 d^2 \log ^2(F)}+\frac {\left (6 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)} \\ & = \frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}-\frac {\left (6 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)} \\ & = -\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.38 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\frac {F^{a+b (c+d x)} \left (-6 f^2+2 b d f (2 e+3 f x) \log (F)-b^2 d^2 \left (e^2+4 e f x+3 f^2 x^2\right ) \log ^2(F)+b^3 d^3 x (e+f x)^2 \log ^3(F)\right )}{b^4 d^4 \log ^4(F)} \]

[In]

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

[Out]

(F^(a + b*(c + d*x))*(-6*f^2 + 2*b*d*f*(2*e + 3*f*x)*Log[F] - b^2*d^2*(e^2 + 4*e*f*x + 3*f^2*x^2)*Log[F]^2 + b
^3*d^3*x*(e + f*x)^2*Log[F]^3))/(b^4*d^4*Log[F]^4)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {\left (\ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+2 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+\ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -3 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-4 \ln \left (F \right )^{2} b^{2} d^{2} e f x -\ln \left (F \right )^{2} b^{2} d^{2} e^{2}+6 \ln \left (F \right ) b d \,f^{2} x +4 e f \ln \left (F \right ) b d -6 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{4} b^{4} d^{4}}\)	\(144\)
risch	\(\frac {\left (\ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+2 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+\ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -3 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-4 \ln \left (F \right )^{2} b^{2} d^{2} e f x -\ln \left (F \right )^{2} b^{2} d^{2} e^{2}+6 \ln \left (F \right ) b d \,f^{2} x +4 e f \ln \left (F \right ) b d -6 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{4} b^{4} d^{4}}\)	\(144\)
meijerg	\(\frac {F^{c b +a} f^{2} \left (6-\frac {\left (-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 b d x \ln \left (F \right )+24\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{4}\right )}{\ln \left (F \right )^{4} b^{4} d^{4}}-\frac {2 F^{c b +a} f e \left (2-\frac {\left (3 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-6 b d x \ln \left (F \right )+6\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{3}\right )}{b^{3} d^{3} \ln \left (F \right )^{3}}+\frac {F^{c b +a} e^{2} \left (1-\frac {\left (-2 b d x \ln \left (F \right )+2\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{2}\right )}{b^{2} d^{2} \ln \left (F \right )^{2}}\)	\(170\)
norman	\(\frac {f^{2} x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{b d \ln \left (F \right )}+\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-4 e f \ln \left (F \right ) b d +6 f^{2}\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} d^{3}}+\frac {f \left (2 \ln \left (F \right ) b d e -3 f \right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} d^{2}}-\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-4 e f \ln \left (F \right ) b d +6 f^{2}\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} d^{4}}\)	\(177\)
parallelrisch	\(\frac {\ln \left (F \right )^{3} x^{3} F^{b d x +c b +a} b^{3} d^{3} f^{2}+2 \ln \left (F \right )^{3} x^{2} F^{b d x +c b +a} b^{3} d^{3} e f +\ln \left (F \right )^{3} x \,F^{b d x +c b +a} b^{3} d^{3} e^{2}-3 \ln \left (F \right )^{2} x^{2} F^{b d x +c b +a} b^{2} d^{2} f^{2}-4 \ln \left (F \right )^{2} x \,F^{b d x +c b +a} b^{2} d^{2} e f -\ln \left (F \right )^{2} F^{b d x +c b +a} b^{2} d^{2} e^{2}+6 \ln \left (F \right ) x \,F^{b d x +c b +a} b d \,f^{2}+4 \ln \left (F \right ) F^{b d x +c b +a} b d e f -6 F^{b d x +c b +a} f^{2}}{\ln \left (F \right )^{4} b^{4} d^{4}}\)	\(232\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.55 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\frac {{\left ({\left (b^{3} d^{3} f^{2} x^{3} + 2 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} - {\left (3 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 6 \, f^{2} + 2 \, {\left (3 \, b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{4} d^{4} \log \left (F\right )^{4}} \]

[In]

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

[Out]

((b^3*d^3*f^2*x^3 + 2*b^3*d^3*e*f*x^2 + b^3*d^3*e^2*x)*log(F)^3 - (3*b^2*d^2*f^2*x^2 + 4*b^2*d^2*e*f*x + b^2*d
^2*e^2)*log(F)^2 - 6*f^2 + 2*(3*b*d*f^2*x + 2*b*d*e*f)*log(F))*F^(b*d*x + b*c + a)/(b^4*d^4*log(F)^4)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.82 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{3} d^{3} e^{2} x \log {\left (F \right )}^{3} + 2 b^{3} d^{3} e f x^{2} \log {\left (F \right )}^{3} + b^{3} d^{3} f^{2} x^{3} \log {\left (F \right )}^{3} - b^{2} d^{2} e^{2} \log {\left (F \right )}^{2} - 4 b^{2} d^{2} e f x \log {\left (F \right )}^{2} - 3 b^{2} d^{2} f^{2} x^{2} \log {\left (F \right )}^{2} + 4 b d e f \log {\left (F \right )} + 6 b d f^{2} x \log {\left (F \right )} - 6 f^{2}\right )}{b^{4} d^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d^{4} \log {\left (F \right )}^{4} \neq 0 \\\frac {e^{2} x^{2}}{2} + \frac {2 e f x^{3}}{3} + \frac {f^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**3*d**3*e**2*x*log(F)**3 + 2*b**3*d**3*e*f*x**2*log(F)**3 + b**3*d**3*f**2*
x**3*log(F)**3 - b**2*d**2*e**2*log(F)**2 - 4*b**2*d**2*e*f*x*log(F)**2 - 3*b**2*d**2*f**2*x**2*log(F)**2 + 4*
b*d*e*f*log(F) + 6*b*d*f**2*x*log(F) - 6*f**2)/(b**4*d**4*log(F)**4), Ne(b**4*d**4*log(F)**4, 0)), (e**2*x**2/
2 + 2*e*f*x**3/3 + f**2*x**4/4, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.81 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\frac {{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} e^{2}}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e f}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{4} d^{4} \log \left (F\right )^{4}} \]

[In]

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

[Out]

(F^(b*c + a)*b*d*x*log(F) - F^(b*c + a))*F^(b*d*x)*e^2/(b^2*d^2*log(F)^2) + 2*(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*f/(b^3*d^3*log(F)^3) + (F^(b*c + a)*b^3*d^3*x^3*lo
g(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)*f^2/(b^4*d
^4*log(F)^4)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 4114, normalized size of antiderivative = 17.00 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(((3*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 2*b^3*d^3*f^2*x^3*log(abs
(F))^3 + 6*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 6*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) + 4*b^3*d^3*e*f*x^2*lo
g(abs(F))^3 + 3*pi^2*b^3*d^3*e^2*x*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*e^2*x*log(abs(F)) + 2*b^3*d^3*e^2*x*log
(abs(F))^3 - 3*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 3*pi^2*b^2*d^2*f^2*x^2 - 6*b^2*d^2*f^2*x^2*log(abs(F))^2 - 4*pi^2
*b^2*d^2*e*f*x*sgn(F) + 4*pi^2*b^2*d^2*e*f*x - 8*b^2*d^2*e*f*x*log(abs(F))^2 - pi^2*b^2*d^2*e^2*sgn(F) + pi^2*
b^2*d^2*e^2 - 2*b^2*d^2*e^2*log(abs(F))^2 + 12*b*d*f^2*x*log(abs(F)) + 8*b*d*e*f*log(abs(F)) - 12*f^2)*(pi^4*b
^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*
log(abs(F))^4)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log
(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F
) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*d^3*f^2*x^3*sgn(F) - 3*pi*b^3*d^3*f^
2*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*f^2*x^3 + 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 2*pi^3*b^3*d^3*e*f*x^
2*sgn(F) - 6*pi*b^3*d^3*e*f*x^2*log(abs(F))^2*sgn(F) - 2*pi^3*b^3*d^3*e*f*x^2 + 6*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 + 6*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 8*pi*b^2*d
^2*e*f*x*log(abs(F))*sgn(F) - 8*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)) - 6*pi*b*d*f^2*x*sgn(F) + 6*pi*b*d*f^2*x - 4*pi*b*d*e*f*sgn(F) + 4*pi*b*d*e*f)*(pi^3*b^4*d
^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)
/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2
*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d
^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2))*cos(-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^3*b^3*d^3*f^2*x^3*sgn(F) - 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*s
gn(F) - pi^3*b^3*d^3*f^2*x^3 + 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 2*pi^3*b^3*d^3*e*f*x^2*sgn(F) - 6*pi*b^3*d
^3*e*f*x^2*log(abs(F))^2*sgn(F) - 2*pi^3*b^3*d^3*e*f*x^2 + 6*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
+ 6*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 8*pi*b^2*d^2*e*f*x*log(abs(F))*
sgn(F) - 8*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)) -
6*pi*b*d*f^2*x*sgn(F) + 6*pi*b*d*f^2*x - 4*pi*b*d*e*f*sgn(F) + 4*pi*b*d*e*f)*(pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^
4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)/((pi^4*b^4
*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*lo
g(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(
F)) + pi*b^4*d^4*log(abs(F))^3)^2) + 4*(3*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*f^2*x^3*log
(abs(F)) + 2*b^3*d^3*f^2*x^3*log(abs(F))^3 + 6*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 6*pi^2*b^3*d^3*e*f*x^
2*log(abs(F)) + 4*b^3*d^3*e*f*x^2*log(abs(F))^3 + 3*pi^2*b^3*d^3*e^2*x*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*e^2
*x*log(abs(F)) + 2*b^3*d^3*e^2*x*log(abs(F))^3 - 3*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 3*pi^2*b^2*d^2*f^2*x^2 - 6*b^
2*d^2*f^2*x^2*log(abs(F))^2 - 4*pi^2*b^2*d^2*e*f*x*sgn(F) + 4*pi^2*b^2*d^2*e*f*x - 8*b^2*d^2*e*f*x*log(abs(F))
^2 - pi^2*b^2*d^2*e^2*sgn(F) + pi^2*b^2*d^2*e^2 - 2*b^2*d^2*e^2*log(abs(F))^2 + 12*b*d*f^2*x*log(abs(F)) + 8*b
*d*e*f*log(abs(F)) - 12*f^2)*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4
*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^
4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b
^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^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*lo
g(abs(F)) + a*log(abs(F))) - 1/2*I*((pi^3*b^3*d^3*f^2*x^3*sgn(F) + 3*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F)
- 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*f^2*x^3 - 3*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 3
*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 2*I*b^3*d^3*f^2*x^3*log(abs(F))^3 + 2*pi^3*b^3*d^3*e*f*x^2*sgn(F) + 6*I*pi
^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 6*pi*b^3*d^3*e*f*x^2*log(abs(F))^2*sgn(F) - 2*pi^3*b^3*d^3*e*f*x^2 - 6
*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) + 6*pi*b^3*d^3*e*f*x^2*log(abs(F))^2 + 4*I*b^3*d^3*e*f*x^2*log(abs(F))^3 +
pi^3*b^3*d^3*e^2*x*sgn(F) + 3*I*pi^2*b^3*d^3*e^2*x*log(abs(F))*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*I*pi^2*b^3*d^3*e^2*x*log(abs(F)) + 3*pi*b^3*d^3*e^2*x*log(abs(F))^2 + 2*I*b^3*d^3*
e^2*x*log(abs(F))^3 - 3*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 6*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 3*I*pi^2*b^2
*d^2*f^2*x^2 - 6*pi*b^2*d^2*f^2*x^2*log(abs(F)) - 6*I*b^2*d^2*f^2*x^2*log(abs(F))^2 - 4*I*pi^2*b^2*d^2*e*f*x*s
gn(F) + 8*pi*b^2*d^2*e*f*x*log(abs(F))*sgn(F) + 4*I*pi^2*b^2*d^2*e*f*x - 8*pi*b^2*d^2*e*f*x*log(abs(F)) - 8*I*
b^2*d^2*e*f*x*log(abs(F))^2 - I*pi^2*b^2*d^2*e^2*sgn(F) + 2*pi*b^2*d^2*e^2*log(abs(F))*sgn(F) + I*pi^2*b^2*d^2
*e^2 - 2*pi*b^2*d^2*e^2*log(abs(F)) - 2*I*b^2*d^2*e^2*log(abs(F))^2 - 6*pi*b*d*f^2*x*sgn(F) + 6*pi*b*d*f^2*x +
12*I*b*d*f^2*x*log(abs(F)) - 4*pi*b*d*e*f*sgn(F) + 4*pi*b*d*e*f + 8*I*b*d*e*f*log(abs(F)) - 12*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)/(pi^
4*b^4*d^4*sgn(F) + 4*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - 4*I*pi*b^4*d^4*
log(abs(F))^3*sgn(F) - pi^4*b^4*d^4 - 4*I*pi^3*b^4*d^4*log(abs(F)) + 6*pi^2*b^4*d^4*log(abs(F))^2 + 4*I*pi*b^4
*d^4*log(abs(F))^3 - 2*b^4*d^4*log(abs(F))^4) + (pi^3*b^3*d^3*f^2*x^3*sgn(F) - 3*I*pi^2*b^3*d^3*f^2*x^3*log(ab
s(F))*sgn(F) - 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*f^2*x^3 + 3*I*pi^2*b^3*d^3*f^2*x^3*log
(abs(F)) + 3*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 2*I*b^3*d^3*f^2*x^3*log(abs(F))^3 + 2*pi^3*b^3*d^3*e*f*x^2*sgn
(F) - 6*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F))*sgn(F) - 6*pi*b^3*d^3*e*f*x^2*log(abs(F))^2*sgn(F) - 2*pi^3*b^3*d^3
*e*f*x^2 + 6*I*pi^2*b^3*d^3*e*f*x^2*log(abs(F)) + 6*pi*b^3*d^3*e*f*x^2*log(abs(F))^2 - 4*I*b^3*d^3*e*f*x^2*log
(abs(F))^3 + pi^3*b^3*d^3*e^2*x*sgn(F) - 3*I*pi^2*b^3*d^3*e^2*x*log(abs(F))*sgn(F) - 3*pi*b^3*d^3*e^2*x*log(ab
s(F))^2*sgn(F) - pi^3*b^3*d^3*e^2*x + 3*I*pi^2*b^3*d^3*e^2*x*log(abs(F)) + 3*pi*b^3*d^3*e^2*x*log(abs(F))^2 -
2*I*b^3*d^3*e^2*x*log(abs(F))^3 + 3*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 6*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) -
3*I*pi^2*b^2*d^2*f^2*x^2 - 6*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 6*I*b^2*d^2*f^2*x^2*log(abs(F))^2 + 4*I*pi^2*b^2
*d^2*e*f*x*sgn(F) + 8*pi*b^2*d^2*e*f*x*log(abs(F))*sgn(F) - 4*I*pi^2*b^2*d^2*e*f*x - 8*pi*b^2*d^2*e*f*x*log(ab
s(F)) + 8*I*b^2*d^2*e*f*x*log(abs(F))^2 + I*pi^2*b^2*d^2*e^2*sgn(F) + 2*pi*b^2*d^2*e^2*log(abs(F))*sgn(F) - I*
pi^2*b^2*d^2*e^2 - 2*pi*b^2*d^2*e^2*log(abs(F)) + 2*I*b^2*d^2*e^2*log(abs(F))^2 - 6*pi*b*d*f^2*x*sgn(F) + 6*pi
*b*d*f^2*x - 12*I*b*d*f^2*x*log(abs(F)) - 4*pi*b*d*e*f*sgn(F) + 4*pi*b*d*e*f - 8*I*b*d*e*f*log(abs(F)) + 12*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)/(pi^4*b^4*d^4*sgn(F) - 4*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) + 4*
I*pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^4*b^4*d^4 + 4*I*pi^3*b^4*d^4*log(abs(F)) + 6*pi^2*b^4*d^4*log(abs(F))^2
- 4*I*pi*b^4*d^4*log(abs(F))^3 - 2*b^4*d^4*log(abs(F))^4))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs
(F)))

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.59 \[ \int F^{a+b (c+d x)} x (e+f x)^2 \, dx=\frac {F^{a+b\,c+b\,d\,x}\,\left (b^3\,d^3\,e^2\,x\,{\ln \left (F\right )}^3+2\,b^3\,d^3\,e\,f\,x^2\,{\ln \left (F\right )}^3+b^3\,d^3\,f^2\,x^3\,{\ln \left (F\right )}^3-b^2\,d^2\,e^2\,{\ln \left (F\right )}^2-4\,b^2\,d^2\,e\,f\,x\,{\ln \left (F\right )}^2-3\,b^2\,d^2\,f^2\,x^2\,{\ln \left (F\right )}^2+4\,b\,d\,e\,f\,\ln \left (F\right )+6\,b\,d\,f^2\,x\,\ln \left (F\right )-6\,f^2\right )}{b^4\,d^4\,{\ln \left (F\right )}^4} \]

[In]

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

[Out]

(F^(a + b*c + b*d*x)*(6*b*d*f^2*x*log(F) - b^2*d^2*e^2*log(F)^2 - 6*f^2 + b^3*d^3*e^2*x*log(F)^3 - 3*b^2*d^2*f
^2*x^2*log(F)^2 + b^3*d^3*f^2*x^3*log(F)^3 + 4*b*d*e*f*log(F) - 4*b^2*d^2*e*f*x*log(F)^2 + 2*b^3*d^3*e*f*x^2*l
og(F)^3))/(b^4*d^4*log(F)^4)