\(\int F^{c (a+b x)} (d+e x)^4 \, dx\) [2]
Optimal result
Integrand size = 17, antiderivative size = 141 \[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^3 F^{c (a+b x)} (d+e x)}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 F^{c (a+b x)} (d+e x)^2}{b^3 c^3 \log ^3(F)}-\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)}
\]
[Out]
24*e^4*F^(c*(b*x+a))/b^5/c^5/ln(F)^5-24*e^3*F^(c*(b*x+a))*(e*x+d)/b^4/c^4/ln(F)^4+12*e^2*F^(c*(b*x+a))*(e*x+d)
^2/b^3/c^3/ln(F)^3-4*e*F^(c*(b*x+a))*(e*x+d)^3/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(e*x+d)^4/b/c/ln(F)
Rubi [A] (verified)
Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number
of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2207, 2225}
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^3 (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 (d+e x)^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {4 e (d+e x)^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}
\]
[In]
Int[F^(c*(a + b*x))*(d + e*x)^4,x]
[Out]
(24*e^4*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (24*e^3*F^(c*(a + b*x))*(d + e*x))/(b^4*c^4*Log[F]^4) + (12*e^2*
F^(c*(a + b*x))*(d + e*x)^2)/(b^3*c^3*Log[F]^3) - (4*e*F^(c*(a + b*x))*(d + e*x)^3)/(b^2*c^2*Log[F]^2) + (F^(c
*(a + b*x))*(d + e*x)^4)/(b*c*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]
Rubi steps \begin{align*}
\text {integral}& = \frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)}-\frac {(4 e) \int F^{c (a+b x)} (d+e x)^3 \, dx}{b c \log (F)} \\ & = -\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)}+\frac {\left (12 e^2\right ) \int F^{c (a+b x)} (d+e x)^2 \, dx}{b^2 c^2 \log ^2(F)} \\ & = \frac {12 e^2 F^{c (a+b x)} (d+e x)^2}{b^3 c^3 \log ^3(F)}-\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)}-\frac {\left (24 e^3\right ) \int F^{c (a+b x)} (d+e x) \, dx}{b^3 c^3 \log ^3(F)} \\ & = -\frac {24 e^3 F^{c (a+b x)} (d+e x)}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 F^{c (a+b x)} (d+e x)^2}{b^3 c^3 \log ^3(F)}-\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)}+\frac {\left (24 e^4\right ) \int F^{c (a+b x)} \, dx}{b^4 c^4 \log ^4(F)} \\ & = \frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^3 F^{c (a+b x)} (d+e x)}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 F^{c (a+b x)} (d+e x)^2}{b^3 c^3 \log ^3(F)}-\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {F^{c (a+b x)} \left (24 e^4-24 b c e^3 (d+e x) \log (F)+12 b^2 c^2 e^2 (d+e x)^2 \log ^2(F)-4 b^3 c^3 e (d+e x)^3 \log ^3(F)+b^4 c^4 (d+e x)^4 \log ^4(F)\right )}{b^5 c^5 \log ^5(F)}
\]
[In]
Integrate[F^(c*(a + b*x))*(d + e*x)^4,x]
[Out]
(F^(c*(a + b*x))*(24*e^4 - 24*b*c*e^3*(d + e*x)*Log[F] + 12*b^2*c^2*e^2*(d + e*x)^2*Log[F]^2 - 4*b^3*c^3*e*(d
+ e*x)^3*Log[F]^3 + b^4*c^4*(d + e*x)^4*Log[F]^4))/(b^5*c^5*Log[F]^5)
Maple [A] (verified)
Time = 0.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.84
| | |
| method | result | size |
| | |
| gosper |
\(\frac {\left (e^{4} x^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 \ln \left (F \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+6 \ln \left (F \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} d^{3} e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} e^{4} x^{3}-12 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e^{2} x -4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{4} x^{2}+24 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{3} x +12 c^{2} b^{2} \ln \left (F \right )^{2} d^{2} e^{2}-24 \ln \left (F \right ) b c \,e^{4} x -24 d \,e^{3} c b \ln \left (F \right )+24 e^{4}\right ) F^{c \left (b x +a \right )}}{c^{5} b^{5} \ln \left (F \right )^{5}}\) |
\(260\) |
| risch |
\(\frac {\left (e^{4} x^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 \ln \left (F \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+6 \ln \left (F \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} d^{3} e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} e^{4} x^{3}-12 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e^{2} x -4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{4} x^{2}+24 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{3} x +12 c^{2} b^{2} \ln \left (F \right )^{2} d^{2} e^{2}-24 \ln \left (F \right ) b c \,e^{4} x -24 d \,e^{3} c b \ln \left (F \right )+24 e^{4}\right ) F^{c \left (b x +a \right )}}{c^{5} b^{5} \ln \left (F \right )^{5}}\) |
\(260\) |
| norman |
\(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 c^{2} b^{2} \ln \left (F \right )^{2} d^{2} e^{2}-24 d \,e^{3} c b \ln \left (F \right )+24 e^{4}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{5} b^{5} \ln \left (F \right )^{5}}+\frac {e^{4} x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c b \ln \left (F \right )}+\frac {4 e \left (c^{3} b^{3} \ln \left (F \right )^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}+\frac {6 e^{2} \left (\ln \left (F \right )^{2} b^{2} c^{2} d^{2}-2 \ln \left (F \right ) b c e d +2 e^{2}\right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {4 e^{3} \left (\ln \left (F \right ) b c d -e \right ) x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{2} b^{2} \ln \left (F \right )^{2}}\) |
\(278\) |
| meijerg |
\(-\frac {F^{c a} e^{4} \left (24-\frac {\left (5 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-120 b c x \ln \left (F \right )+120\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{5}\right )}{c^{5} b^{5} \ln \left (F \right )^{5}}+\frac {4 F^{c a} e^{3} d \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{c^{4} b^{4} \ln \left (F \right )^{4}}-\frac {6 F^{c a} e^{2} d^{2} \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {4 F^{c a} e \,d^{3} \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {F^{c a} d^{4} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{c b \ln \left (F \right )}\) |
\(288\) |
| parallelrisch |
\(\frac {x^{4} F^{c \left (b x +a \right )} e^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 \ln \left (F \right )^{4} x^{3} F^{c \left (b x +a \right )} b^{4} c^{4} d \,e^{3}+6 \ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} d^{2} e^{2}+4 \ln \left (F \right )^{4} x \,F^{c \left (b x +a \right )} b^{4} c^{4} d^{3} e +\ln \left (F \right )^{4} F^{c \left (b x +a \right )} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} x^{3} F^{c \left (b x +a \right )} b^{3} c^{3} e^{4}-12 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} d \,e^{3}-12 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d^{2} e^{2}-4 \ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{4}+24 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} d \,e^{3}+12 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{4}-24 \ln \left (F \right ) F^{c \left (b x +a \right )} b c d \,e^{3}+24 F^{c \left (b x +a \right )} e^{4}}{c^{5} b^{5} \ln \left (F \right )^{5}}\) |
\(386\) |
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[In]
int(F^(c*(b*x+a))*(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
(e^4*x^4*c^4*b^4*ln(F)^4+4*ln(F)^4*b^4*c^4*d*e^3*x^3+6*ln(F)^4*b^4*c^4*d^2*e^2*x^2+4*ln(F)^4*b^4*c^4*d^3*e*x+l
n(F)^4*b^4*c^4*d^4-4*ln(F)^3*b^3*c^3*e^4*x^3-12*ln(F)^3*b^3*c^3*d*e^3*x^2-12*ln(F)^3*b^3*c^3*d^2*e^2*x-4*ln(F)
^3*b^3*c^3*d^3*e+12*ln(F)^2*b^2*c^2*e^4*x^2+24*ln(F)^2*b^2*c^2*d*e^3*x+12*c^2*b^2*ln(F)^2*d^2*e^2-24*ln(F)*b*c
*e^4*x-24*d*e^3*c*b*ln(F)+24*e^4)*F^(c*(b*x+a))/c^5/b^5/ln(F)^5
Fricas [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.61
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {{\left ({\left (b^{4} c^{4} e^{4} x^{4} + 4 \, b^{4} c^{4} d e^{3} x^{3} + 6 \, b^{4} c^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} c^{4} d^{3} e x + b^{4} c^{4} d^{4}\right )} \log \left (F\right )^{4} + 24 \, e^{4} - 4 \, {\left (b^{3} c^{3} e^{4} x^{3} + 3 \, b^{3} c^{3} d e^{3} x^{2} + 3 \, b^{3} c^{3} d^{2} e^{2} x + b^{3} c^{3} d^{3} e\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} c^{2} e^{4} x^{2} + 2 \, b^{2} c^{2} d e^{3} x + b^{2} c^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 24 \, {\left (b c e^{4} x + b c d e^{3}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}}
\]
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^4,x, algorithm="fricas")
[Out]
((b^4*c^4*e^4*x^4 + 4*b^4*c^4*d*e^3*x^3 + 6*b^4*c^4*d^2*e^2*x^2 + 4*b^4*c^4*d^3*e*x + b^4*c^4*d^4)*log(F)^4 +
24*e^4 - 4*(b^3*c^3*e^4*x^3 + 3*b^3*c^3*d*e^3*x^2 + 3*b^3*c^3*d^2*e^2*x + b^3*c^3*d^3*e)*log(F)^3 + 12*(b^2*c^
2*e^4*x^2 + 2*b^2*c^2*d*e^3*x + b^2*c^2*d^2*e^2)*log(F)^2 - 24*(b*c*e^4*x + b*c*d*e^3)*log(F))*F^(b*c*x + a*c)
/(b^5*c^5*log(F)^5)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (139) = 278\).
Time = 0.11 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.48
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{4} \log {\left (F \right )}^{4} + 4 b^{4} c^{4} d^{3} e x \log {\left (F \right )}^{4} + 6 b^{4} c^{4} d^{2} e^{2} x^{2} \log {\left (F \right )}^{4} + 4 b^{4} c^{4} d e^{3} x^{3} \log {\left (F \right )}^{4} + b^{4} c^{4} e^{4} x^{4} \log {\left (F \right )}^{4} - 4 b^{3} c^{3} d^{3} e \log {\left (F \right )}^{3} - 12 b^{3} c^{3} d^{2} e^{2} x \log {\left (F \right )}^{3} - 12 b^{3} c^{3} d e^{3} x^{2} \log {\left (F \right )}^{3} - 4 b^{3} c^{3} e^{4} x^{3} \log {\left (F \right )}^{3} + 12 b^{2} c^{2} d^{2} e^{2} \log {\left (F \right )}^{2} + 24 b^{2} c^{2} d e^{3} x \log {\left (F \right )}^{2} + 12 b^{2} c^{2} e^{4} x^{2} \log {\left (F \right )}^{2} - 24 b c d e^{3} \log {\left (F \right )} - 24 b c e^{4} x \log {\left (F \right )} + 24 e^{4}\right )}{b^{5} c^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} c^{5} \log {\left (F \right )}^{5} \neq 0 \\d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5} & \text {otherwise} \end {cases}
\]
[In]
integrate(F**(c*(b*x+a))*(e*x+d)**4,x)
[Out]
Piecewise((F**(c*(a + b*x))*(b**4*c**4*d**4*log(F)**4 + 4*b**4*c**4*d**3*e*x*log(F)**4 + 6*b**4*c**4*d**2*e**2
*x**2*log(F)**4 + 4*b**4*c**4*d*e**3*x**3*log(F)**4 + b**4*c**4*e**4*x**4*log(F)**4 - 4*b**3*c**3*d**3*e*log(F
)**3 - 12*b**3*c**3*d**2*e**2*x*log(F)**3 - 12*b**3*c**3*d*e**3*x**2*log(F)**3 - 4*b**3*c**3*e**4*x**3*log(F)*
*3 + 12*b**2*c**2*d**2*e**2*log(F)**2 + 24*b**2*c**2*d*e**3*x*log(F)**2 + 12*b**2*c**2*e**4*x**2*log(F)**2 - 2
4*b*c*d*e**3*log(F) - 24*b*c*e**4*x*log(F) + 24*e**4)/(b**5*c**5*log(F)**5), Ne(b**5*c**5*log(F)**5, 0)), (d**
4*x + 2*d**3*e*x**2 + 2*d**2*e**2*x**3 + d*e**3*x**4 + e**4*x**5/5, True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (141) = 282\).
Time = 0.20 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.19
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {F^{b c x + a c} d^{4}}{b c \log \left (F\right )} + \frac {4 \, {\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{3} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {6 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d^{2} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {4 \, {\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} d e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} e^{4}}{b^{5} c^{5} \log \left (F\right )^{5}}
\]
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^4,x, algorithm="maxima")
[Out]
F^(b*c*x + a*c)*d^4/(b*c*log(F)) + 4*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*d^3*e/(b^2*c^2*log(F)^2) + 6*(
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^2*e^2/(b^3*c^3*log(F)^3) + 4*(F
^(a*c)*b^3*c^3*x^3*log(F)^3 - 3*F^(a*c)*b^2*c^2*x^2*log(F)^2 + 6*F^(a*c)*b*c*x*log(F) - 6*F^(a*c))*F^(b*c*x)*d
*e^3/(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)*e^4/(b^5*c^5*log(F)^5)
Giac [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 8802, normalized size of antiderivative = 62.43
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\text {Too large to display}
\]
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^4,x, algorithm="giac")
[Out]
-((4*(pi^3*b^4*c^4*e^4*x^4*log(abs(F))*sgn(F) - pi*b^4*c^4*e^4*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*e^4*x^4
*log(abs(F)) + pi*b^4*c^4*e^4*x^4*log(abs(F))^3 + 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*d
*e^3*x^3*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F)) + 4*pi*b^4*c^4*d*e^3*x^3*log(abs(F))^3 +
6*pi^3*b^4*c^4*d^2*e^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3*sgn(F) - 6*pi^3*b^4*c^4
*d^2*e^2*x^2*log(abs(F)) + 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3 + 4*pi^3*b^4*c^4*d^3*e*x*log(abs(F))*sgn(F)
- 4*pi*b^4*c^4*d^3*e*x*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*d^3*e*x*log(abs(F)) + 4*pi*b^4*c^4*d^3*e*x*log(ab
s(F))^3 - pi^3*b^3*c^3*e^4*x^3*sgn(F) + pi^3*b^4*c^4*d^4*log(abs(F))*sgn(F) + 3*pi*b^3*c^3*e^4*x^3*log(abs(F))
^2*sgn(F) - pi*b^4*c^4*d^4*log(abs(F))^3*sgn(F) + pi^3*b^3*c^3*e^4*x^3 - pi^3*b^4*c^4*d^4*log(abs(F)) - 3*pi*b
^3*c^3*e^4*x^3*log(abs(F))^2 + pi*b^4*c^4*d^4*log(abs(F))^3 - 3*pi^3*b^3*c^3*d*e^3*x^2*sgn(F) + 9*pi*b^3*c^3*d
*e^3*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*d*e^3*x^2 - 9*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2 - 3*pi^3*b^3*c
^3*d^2*e^2*x*sgn(F) + 9*pi*b^3*c^3*d^2*e^2*x*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*d^2*e^2*x - 9*pi*b^3*c^3*d^
2*e^2*x*log(abs(F))^2 - pi^3*b^3*c^3*d^3*e*sgn(F) + 3*pi*b^3*c^3*d^3*e*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*d^3
*e - 3*pi*b^3*c^3*d^3*e*log(abs(F))^2 - 6*pi*b^2*c^2*e^4*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*c^2*e^4*x^2*log(abs
(F)) - 12*pi*b^2*c^2*d*e^3*x*log(abs(F))*sgn(F) + 12*pi*b^2*c^2*d*e^3*x*log(abs(F)) - 6*pi*b^2*c^2*d^2*e^2*log
(abs(F))*sgn(F) + 6*pi*b^2*c^2*d^2*e^2*log(abs(F)) + 6*pi*b*c*e^4*x*sgn(F) - 6*pi*b*c*e^4*x + 6*pi*b*c*d*e^3*s
gn(F) - 6*pi*b*c*d*e^3)*(pi^5*b^5*c^5*sgn(F) - 10*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))
^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*b^5*c^5*log(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)/((pi^5*b^5*c^5*sgn(F) -
10*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*b^5*c^5*log
(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)^2 + (5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*log(abs(F))^
3*sgn(F) - 5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))^5)^2) - (pi^4*b^
4*c^4*e^4*x^4*sgn(F) - 6*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*e^4*x^4 + 6*pi^2*b^4*c^4*e^4
*x^4*log(abs(F))^2 - 2*b^4*c^4*e^4*x^4*log(abs(F))^4 + 4*pi^4*b^4*c^4*d*e^3*x^3*sgn(F) - 24*pi^2*b^4*c^4*d*e^3
*x^3*log(abs(F))^2*sgn(F) - 4*pi^4*b^4*c^4*d*e^3*x^3 + 24*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^2 - 8*b^4*c^4*d*e
^3*x^3*log(abs(F))^4 + 6*pi^4*b^4*c^4*d^2*e^2*x^2*sgn(F) - 36*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2*sgn(F) -
6*pi^4*b^4*c^4*d^2*e^2*x^2 + 36*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2 - 12*b^4*c^4*d^2*e^2*x^2*log(abs(F))^4
+ 4*pi^4*b^4*c^4*d^3*e*x*sgn(F) - 24*pi^2*b^4*c^4*d^3*e*x*log(abs(F))^2*sgn(F) - 4*pi^4*b^4*c^4*d^3*e*x + 24*p
i^2*b^4*c^4*d^3*e*x*log(abs(F))^2 - 8*b^4*c^4*d^3*e*x*log(abs(F))^4 + pi^4*b^4*c^4*d^4*sgn(F) + 12*pi^2*b^3*c^
3*e^4*x^3*log(abs(F))*sgn(F) - 6*pi^2*b^4*c^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*d^4 - 12*pi^2*b^3*c^3*e^
4*x^3*log(abs(F)) + 6*pi^2*b^4*c^4*d^4*log(abs(F))^2 + 8*b^3*c^3*e^4*x^3*log(abs(F))^3 - 2*b^4*c^4*d^4*log(abs
(F))^4 + 36*pi^2*b^3*c^3*d*e^3*x^2*log(abs(F))*sgn(F) - 36*pi^2*b^3*c^3*d*e^3*x^2*log(abs(F)) + 24*b^3*c^3*d*e
^3*x^2*log(abs(F))^3 + 36*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F))*sgn(F) - 36*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F)) +
24*b^3*c^3*d^2*e^2*x*log(abs(F))^3 + 12*pi^2*b^3*c^3*d^3*e*log(abs(F))*sgn(F) - 12*pi^2*b^3*c^3*d^3*e*log(abs(
F)) + 8*b^3*c^3*d^3*e*log(abs(F))^3 - 12*pi^2*b^2*c^2*e^4*x^2*sgn(F) + 12*pi^2*b^2*c^2*e^4*x^2 - 24*b^2*c^2*e^
4*x^2*log(abs(F))^2 - 24*pi^2*b^2*c^2*d*e^3*x*sgn(F) + 24*pi^2*b^2*c^2*d*e^3*x - 48*b^2*c^2*d*e^3*x*log(abs(F)
)^2 - 12*pi^2*b^2*c^2*d^2*e^2*sgn(F) + 12*pi^2*b^2*c^2*d^2*e^2 - 24*b^2*c^2*d^2*e^2*log(abs(F))^2 + 48*b*c*e^4
*x*log(abs(F)) + 48*b*c*d*e^3*log(abs(F)) - 48*e^4)*(5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*log(a
bs(F))^3*sgn(F) - 5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))^5)/((pi^5
*b^5*c^5*sgn(F) - 10*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10
*pi^3*b^5*c^5*log(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)^2 + (5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5
*c^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))
^5)^2))*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c) - ((pi^4*b^4*c^4*e^4*x^4*sgn
(F) - 6*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*e^4*x^4 + 6*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^
2 - 2*b^4*c^4*e^4*x^4*log(abs(F))^4 + 4*pi^4*b^4*c^4*d*e^3*x^3*sgn(F) - 24*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^
2*sgn(F) - 4*pi^4*b^4*c^4*d*e^3*x^3 + 24*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^2 - 8*b^4*c^4*d*e^3*x^3*log(abs(F)
)^4 + 6*pi^4*b^4*c^4*d^2*e^2*x^2*sgn(F) - 36*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2*sgn(F) - 6*pi^4*b^4*c^4*d^
2*e^2*x^2 + 36*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2 - 12*b^4*c^4*d^2*e^2*x^2*log(abs(F))^4 + 4*pi^4*b^4*c^4*
d^3*e*x*sgn(F) - 24*pi^2*b^4*c^4*d^3*e*x*log(abs(F))^2*sgn(F) - 4*pi^4*b^4*c^4*d^3*e*x + 24*pi^2*b^4*c^4*d^3*e
*x*log(abs(F))^2 - 8*b^4*c^4*d^3*e*x*log(abs(F))^4 + pi^4*b^4*c^4*d^4*sgn(F) + 12*pi^2*b^3*c^3*e^4*x^3*log(abs
(F))*sgn(F) - 6*pi^2*b^4*c^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*d^4 - 12*pi^2*b^3*c^3*e^4*x^3*log(abs(F))
+ 6*pi^2*b^4*c^4*d^4*log(abs(F))^2 + 8*b^3*c^3*e^4*x^3*log(abs(F))^3 - 2*b^4*c^4*d^4*log(abs(F))^4 + 36*pi^2*
b^3*c^3*d*e^3*x^2*log(abs(F))*sgn(F) - 36*pi^2*b^3*c^3*d*e^3*x^2*log(abs(F)) + 24*b^3*c^3*d*e^3*x^2*log(abs(F)
)^3 + 36*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F))*sgn(F) - 36*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F)) + 24*b^3*c^3*d^2*e^
2*x*log(abs(F))^3 + 12*pi^2*b^3*c^3*d^3*e*log(abs(F))*sgn(F) - 12*pi^2*b^3*c^3*d^3*e*log(abs(F)) + 8*b^3*c^3*d
^3*e*log(abs(F))^3 - 12*pi^2*b^2*c^2*e^4*x^2*sgn(F) + 12*pi^2*b^2*c^2*e^4*x^2 - 24*b^2*c^2*e^4*x^2*log(abs(F))
^2 - 24*pi^2*b^2*c^2*d*e^3*x*sgn(F) + 24*pi^2*b^2*c^2*d*e^3*x - 48*b^2*c^2*d*e^3*x*log(abs(F))^2 - 12*pi^2*b^2
*c^2*d^2*e^2*sgn(F) + 12*pi^2*b^2*c^2*d^2*e^2 - 24*b^2*c^2*d^2*e^2*log(abs(F))^2 + 48*b*c*e^4*x*log(abs(F)) +
48*b*c*d*e^3*log(abs(F)) - 48*e^4)*(pi^5*b^5*c^5*sgn(F) - 10*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*
log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*b^5*c^5*log(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)/((pi^5*b^5*c
^5*sgn(F) - 10*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*
b^5*c^5*log(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)^2 + (5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*l
og(abs(F))^3*sgn(F) - 5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))^5)^2)
+ 4*(pi^3*b^4*c^4*e^4*x^4*log(abs(F))*sgn(F) - pi*b^4*c^4*e^4*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*e^4*x^4
*log(abs(F)) + pi*b^4*c^4*e^4*x^4*log(abs(F))^3 + 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*d
*e^3*x^3*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F)) + 4*pi*b^4*c^4*d*e^3*x^3*log(abs(F))^3 +
6*pi^3*b^4*c^4*d^2*e^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3*sgn(F) - 6*pi^3*b^4*c^4
*d^2*e^2*x^2*log(abs(F)) + 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3 + 4*pi^3*b^4*c^4*d^3*e*x*log(abs(F))*sgn(F)
- 4*pi*b^4*c^4*d^3*e*x*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*d^3*e*x*log(abs(F)) + 4*pi*b^4*c^4*d^3*e*x*log(ab
s(F))^3 - pi^3*b^3*c^3*e^4*x^3*sgn(F) + pi^3*b^4*c^4*d^4*log(abs(F))*sgn(F) + 3*pi*b^3*c^3*e^4*x^3*log(abs(F))
^2*sgn(F) - pi*b^4*c^4*d^4*log(abs(F))^3*sgn(F) + pi^3*b^3*c^3*e^4*x^3 - pi^3*b^4*c^4*d^4*log(abs(F)) - 3*pi*b
^3*c^3*e^4*x^3*log(abs(F))^2 + pi*b^4*c^4*d^4*log(abs(F))^3 - 3*pi^3*b^3*c^3*d*e^3*x^2*sgn(F) + 9*pi*b^3*c^3*d
*e^3*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*d*e^3*x^2 - 9*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2 - 3*pi^3*b^3*c
^3*d^2*e^2*x*sgn(F) + 9*pi*b^3*c^3*d^2*e^2*x*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*d^2*e^2*x - 9*pi*b^3*c^3*d^
2*e^2*x*log(abs(F))^2 - pi^3*b^3*c^3*d^3*e*sgn(F) + 3*pi*b^3*c^3*d^3*e*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*d^3
*e - 3*pi*b^3*c^3*d^3*e*log(abs(F))^2 - 6*pi*b^2*c^2*e^4*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*c^2*e^4*x^2*log(abs
(F)) - 12*pi*b^2*c^2*d*e^3*x*log(abs(F))*sgn(F) + 12*pi*b^2*c^2*d*e^3*x*log(abs(F)) - 6*pi*b^2*c^2*d^2*e^2*log
(abs(F))*sgn(F) + 6*pi*b^2*c^2*d^2*e^2*log(abs(F)) + 6*pi*b*c*e^4*x*sgn(F) - 6*pi*b*c*e^4*x + 6*pi*b*c*d*e^3*s
gn(F) - 6*pi*b*c*d*e^3)*(5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5
*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))^5)/((pi^5*b^5*c^5*sgn(F) - 10*pi^3*b^
5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*b^5*c^5*log(abs(F))^2
- 5*pi*b^5*c^5*log(abs(F))^4)^2 + (5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*log(abs(F))^3*sgn(F) -
5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*log(abs(F))^5)^2))*sin(-1/2*pi*b*c*x*sg
n(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 8*I*((I*pi^4*
b^4*c^4*e^4*x^4*sgn(F) - 4*pi^3*b^4*c^4*e^4*x^4*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^2*sg
n(F) + 4*pi*b^4*c^4*e^4*x^4*log(abs(F))^3*sgn(F) - I*pi^4*b^4*c^4*e^4*x^4 + 4*pi^3*b^4*c^4*e^4*x^4*log(abs(F))
+ 6*I*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^2 - 4*pi*b^4*c^4*e^4*x^4*log(abs(F))^3 - 2*I*b^4*c^4*e^4*x^4*log(abs(F
))^4 + 4*I*pi^4*b^4*c^4*d*e^3*x^3*sgn(F) - 16*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F))*sgn(F) - 24*I*pi^2*b^4*c^4*d*
e^3*x^3*log(abs(F))^2*sgn(F) + 16*pi*b^4*c^4*d*e^3*x^3*log(abs(F))^3*sgn(F) - 4*I*pi^4*b^4*c^4*d*e^3*x^3 + 16*
pi^3*b^4*c^4*d*e^3*x^3*log(abs(F)) + 24*I*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^2 - 16*pi*b^4*c^4*d*e^3*x^3*log(a
bs(F))^3 - 8*I*b^4*c^4*d*e^3*x^3*log(abs(F))^4 + 6*I*pi^4*b^4*c^4*d^2*e^2*x^2*sgn(F) - 24*pi^3*b^4*c^4*d^2*e^2
*x^2*log(abs(F))*sgn(F) - 36*I*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2*sgn(F) + 24*pi*b^4*c^4*d^2*e^2*x^2*log(a
bs(F))^3*sgn(F) - 6*I*pi^4*b^4*c^4*d^2*e^2*x^2 + 24*pi^3*b^4*c^4*d^2*e^2*x^2*log(abs(F)) + 36*I*pi^2*b^4*c^4*d
^2*e^2*x^2*log(abs(F))^2 - 24*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3 - 12*I*b^4*c^4*d^2*e^2*x^2*log(abs(F))^4 +
4*I*pi^4*b^4*c^4*d^3*e*x*sgn(F) - 16*pi^3*b^4*c^4*d^3*e*x*log(abs(F))*sgn(F) - 24*I*pi^2*b^4*c^4*d^3*e*x*log(a
bs(F))^2*sgn(F) + 16*pi*b^4*c^4*d^3*e*x*log(abs(F))^3*sgn(F) - 4*I*pi^4*b^4*c^4*d^3*e*x + 16*pi^3*b^4*c^4*d^3*
e*x*log(abs(F)) + 24*I*pi^2*b^4*c^4*d^3*e*x*log(abs(F))^2 - 16*pi*b^4*c^4*d^3*e*x*log(abs(F))^3 - 8*I*b^4*c^4*
d^3*e*x*log(abs(F))^4 + I*pi^4*b^4*c^4*d^4*sgn(F) + 4*pi^3*b^3*c^3*e^4*x^3*sgn(F) - 4*pi^3*b^4*c^4*d^4*log(abs
(F))*sgn(F) + 12*I*pi^2*b^3*c^3*e^4*x^3*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*c^4*d^4*log(abs(F))^2*sgn(F) - 12*pi
*b^3*c^3*e^4*x^3*log(abs(F))^2*sgn(F) + 4*pi*b^4*c^4*d^4*log(abs(F))^3*sgn(F) - I*pi^4*b^4*c^4*d^4 - 4*pi^3*b^
3*c^3*e^4*x^3 + 4*pi^3*b^4*c^4*d^4*log(abs(F)) - 12*I*pi^2*b^3*c^3*e^4*x^3*log(abs(F)) + 6*I*pi^2*b^4*c^4*d^4*
log(abs(F))^2 + 12*pi*b^3*c^3*e^4*x^3*log(abs(F))^2 - 4*pi*b^4*c^4*d^4*log(abs(F))^3 + 8*I*b^3*c^3*e^4*x^3*log
(abs(F))^3 - 2*I*b^4*c^4*d^4*log(abs(F))^4 + 12*pi^3*b^3*c^3*d*e^3*x^2*sgn(F) + 36*I*pi^2*b^3*c^3*d*e^3*x^2*lo
g(abs(F))*sgn(F) - 36*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2*sgn(F) - 12*pi^3*b^3*c^3*d*e^3*x^2 - 36*I*pi^2*b^3*c^
3*d*e^3*x^2*log(abs(F)) + 36*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2 + 24*I*b^3*c^3*d*e^3*x^2*log(abs(F))^3 + 12*pi
^3*b^3*c^3*d^2*e^2*x*sgn(F) + 36*I*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F))*sgn(F) - 36*pi*b^3*c^3*d^2*e^2*x*log(abs
(F))^2*sgn(F) - 12*pi^3*b^3*c^3*d^2*e^2*x - 36*I*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F)) + 36*pi*b^3*c^3*d^2*e^2*x*
log(abs(F))^2 + 24*I*b^3*c^3*d^2*e^2*x*log(abs(F))^3 + 4*pi^3*b^3*c^3*d^3*e*sgn(F) + 12*I*pi^2*b^3*c^3*d^3*e*l
og(abs(F))*sgn(F) - 12*pi*b^3*c^3*d^3*e*log(abs(F))^2*sgn(F) - 4*pi^3*b^3*c^3*d^3*e - 12*I*pi^2*b^3*c^3*d^3*e*
log(abs(F)) + 12*pi*b^3*c^3*d^3*e*log(abs(F))^2 + 8*I*b^3*c^3*d^3*e*log(abs(F))^3 - 12*I*pi^2*b^2*c^2*e^4*x^2*
sgn(F) + 24*pi*b^2*c^2*e^4*x^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2*c^2*e^4*x^2 - 24*pi*b^2*c^2*e^4*x^2*log(abs(
F)) - 24*I*b^2*c^2*e^4*x^2*log(abs(F))^2 - 24*I*pi^2*b^2*c^2*d*e^3*x*sgn(F) + 48*pi*b^2*c^2*d*e^3*x*log(abs(F)
)*sgn(F) + 24*I*pi^2*b^2*c^2*d*e^3*x - 48*pi*b^2*c^2*d*e^3*x*log(abs(F)) - 48*I*b^2*c^2*d*e^3*x*log(abs(F))^2
- 12*I*pi^2*b^2*c^2*d^2*e^2*sgn(F) + 24*pi*b^2*c^2*d^2*e^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2*c^2*d^2*e^2 - 24
*pi*b^2*c^2*d^2*e^2*log(abs(F)) - 24*I*b^2*c^2*d^2*e^2*log(abs(F))^2 - 24*pi*b*c*e^4*x*sgn(F) + 24*pi*b*c*e^4*
x + 48*I*b*c*e^4*x*log(abs(F)) - 24*pi*b*c*d*e^3*sgn(F) + 24*pi*b*c*d*e^3 + 48*I*b*c*d*e^3*log(abs(F)) - 48*I*
e^4)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(16*I*pi^5*b^5*c^5*sgn(F)
- 80*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 160*I*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 160*pi^2*b^5*c^5*log(abs(F))
^3*sgn(F) + 80*I*pi*b^5*c^5*log(abs(F))^4*sgn(F) - 16*I*pi^5*b^5*c^5 + 80*pi^4*b^5*c^5*log(abs(F)) + 160*I*pi^
3*b^5*c^5*log(abs(F))^2 - 160*pi^2*b^5*c^5*log(abs(F))^3 - 80*I*pi*b^5*c^5*log(abs(F))^4 + 32*b^5*c^5*log(abs(
F))^5) - (I*pi^4*b^4*c^4*e^4*x^4*sgn(F) + 4*pi^3*b^4*c^4*e^4*x^4*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*c^4*e^4*x^4
*log(abs(F))^2*sgn(F) - 4*pi*b^4*c^4*e^4*x^4*log(abs(F))^3*sgn(F) - I*pi^4*b^4*c^4*e^4*x^4 - 4*pi^3*b^4*c^4*e^
4*x^4*log(abs(F)) + 6*I*pi^2*b^4*c^4*e^4*x^4*log(abs(F))^2 + 4*pi*b^4*c^4*e^4*x^4*log(abs(F))^3 - 2*I*b^4*c^4*
e^4*x^4*log(abs(F))^4 + 4*I*pi^4*b^4*c^4*d*e^3*x^3*sgn(F) + 16*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F))*sgn(F) - 24*
I*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^2*sgn(F) - 16*pi*b^4*c^4*d*e^3*x^3*log(abs(F))^3*sgn(F) - 4*I*pi^4*b^4*c^
4*d*e^3*x^3 - 16*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F)) + 24*I*pi^2*b^4*c^4*d*e^3*x^3*log(abs(F))^2 + 16*pi*b^4*c^
4*d*e^3*x^3*log(abs(F))^3 - 8*I*b^4*c^4*d*e^3*x^3*log(abs(F))^4 + 6*I*pi^4*b^4*c^4*d^2*e^2*x^2*sgn(F) + 24*pi^
3*b^4*c^4*d^2*e^2*x^2*log(abs(F))*sgn(F) - 36*I*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2*sgn(F) - 24*pi*b^4*c^4*
d^2*e^2*x^2*log(abs(F))^3*sgn(F) - 6*I*pi^4*b^4*c^4*d^2*e^2*x^2 - 24*pi^3*b^4*c^4*d^2*e^2*x^2*log(abs(F)) + 36
*I*pi^2*b^4*c^4*d^2*e^2*x^2*log(abs(F))^2 + 24*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3 - 12*I*b^4*c^4*d^2*e^2*x^2
*log(abs(F))^4 + 4*I*pi^4*b^4*c^4*d^3*e*x*sgn(F) + 16*pi^3*b^4*c^4*d^3*e*x*log(abs(F))*sgn(F) - 24*I*pi^2*b^4*
c^4*d^3*e*x*log(abs(F))^2*sgn(F) - 16*pi*b^4*c^4*d^3*e*x*log(abs(F))^3*sgn(F) - 4*I*pi^4*b^4*c^4*d^3*e*x - 16*
pi^3*b^4*c^4*d^3*e*x*log(abs(F)) + 24*I*pi^2*b^4*c^4*d^3*e*x*log(abs(F))^2 + 16*pi*b^4*c^4*d^3*e*x*log(abs(F))
^3 - 8*I*b^4*c^4*d^3*e*x*log(abs(F))^4 + I*pi^4*b^4*c^4*d^4*sgn(F) - 4*pi^3*b^3*c^3*e^4*x^3*sgn(F) + 4*pi^3*b^
4*c^4*d^4*log(abs(F))*sgn(F) + 12*I*pi^2*b^3*c^3*e^4*x^3*log(abs(F))*sgn(F) - 6*I*pi^2*b^4*c^4*d^4*log(abs(F))
^2*sgn(F) + 12*pi*b^3*c^3*e^4*x^3*log(abs(F))^2*sgn(F) - 4*pi*b^4*c^4*d^4*log(abs(F))^3*sgn(F) - I*pi^4*b^4*c^
4*d^4 + 4*pi^3*b^3*c^3*e^4*x^3 - 4*pi^3*b^4*c^4*d^4*log(abs(F)) - 12*I*pi^2*b^3*c^3*e^4*x^3*log(abs(F)) + 6*I*
pi^2*b^4*c^4*d^4*log(abs(F))^2 - 12*pi*b^3*c^3*e^4*x^3*log(abs(F))^2 + 4*pi*b^4*c^4*d^4*log(abs(F))^3 + 8*I*b^
3*c^3*e^4*x^3*log(abs(F))^3 - 2*I*b^4*c^4*d^4*log(abs(F))^4 - 12*pi^3*b^3*c^3*d*e^3*x^2*sgn(F) + 36*I*pi^2*b^3
*c^3*d*e^3*x^2*log(abs(F))*sgn(F) + 36*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2*sgn(F) + 12*pi^3*b^3*c^3*d*e^3*x^2 -
36*I*pi^2*b^3*c^3*d*e^3*x^2*log(abs(F)) - 36*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2 + 24*I*b^3*c^3*d*e^3*x^2*log(
abs(F))^3 - 12*pi^3*b^3*c^3*d^2*e^2*x*sgn(F) + 36*I*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F))*sgn(F) + 36*pi*b^3*c^3*
d^2*e^2*x*log(abs(F))^2*sgn(F) + 12*pi^3*b^3*c^3*d^2*e^2*x - 36*I*pi^2*b^3*c^3*d^2*e^2*x*log(abs(F)) - 36*pi*b
^3*c^3*d^2*e^2*x*log(abs(F))^2 + 24*I*b^3*c^3*d^2*e^2*x*log(abs(F))^3 - 4*pi^3*b^3*c^3*d^3*e*sgn(F) + 12*I*pi^
2*b^3*c^3*d^3*e*log(abs(F))*sgn(F) + 12*pi*b^3*c^3*d^3*e*log(abs(F))^2*sgn(F) + 4*pi^3*b^3*c^3*d^3*e - 12*I*pi
^2*b^3*c^3*d^3*e*log(abs(F)) - 12*pi*b^3*c^3*d^3*e*log(abs(F))^2 + 8*I*b^3*c^3*d^3*e*log(abs(F))^3 - 12*I*pi^2
*b^2*c^2*e^4*x^2*sgn(F) - 24*pi*b^2*c^2*e^4*x^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2*c^2*e^4*x^2 + 24*pi*b^2*c^2
*e^4*x^2*log(abs(F)) - 24*I*b^2*c^2*e^4*x^2*log(abs(F))^2 - 24*I*pi^2*b^2*c^2*d*e^3*x*sgn(F) - 48*pi*b^2*c^2*d
*e^3*x*log(abs(F))*sgn(F) + 24*I*pi^2*b^2*c^2*d*e^3*x + 48*pi*b^2*c^2*d*e^3*x*log(abs(F)) - 48*I*b^2*c^2*d*e^3
*x*log(abs(F))^2 - 12*I*pi^2*b^2*c^2*d^2*e^2*sgn(F) - 24*pi*b^2*c^2*d^2*e^2*log(abs(F))*sgn(F) + 12*I*pi^2*b^2
*c^2*d^2*e^2 + 24*pi*b^2*c^2*d^2*e^2*log(abs(F)) - 24*I*b^2*c^2*d^2*e^2*log(abs(F))^2 + 24*pi*b*c*e^4*x*sgn(F)
- 24*pi*b*c*e^4*x + 48*I*b*c*e^4*x*log(abs(F)) + 24*pi*b*c*d*e^3*sgn(F) - 24*pi*b*c*d*e^3 + 48*I*b*c*d*e^3*lo
g(abs(F)) - 48*I*e^4)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-16*I*
pi^5*b^5*c^5*sgn(F) - 80*pi^4*b^5*c^5*log(abs(F))*sgn(F) + 160*I*pi^3*b^5*c^5*log(abs(F))^2*sgn(F) + 160*pi^2*
b^5*c^5*log(abs(F))^3*sgn(F) - 80*I*pi*b^5*c^5*log(abs(F))^4*sgn(F) + 16*I*pi^5*b^5*c^5 + 80*pi^4*b^5*c^5*log(
abs(F)) - 160*I*pi^3*b^5*c^5*log(abs(F))^2 - 160*pi^2*b^5*c^5*log(abs(F))^3 + 80*I*pi*b^5*c^5*log(abs(F))^4 +
32*b^5*c^5*log(abs(F))^5))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)))
Mupad [B] (verification not implemented)
Time = 0.17 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.84
\[
\int F^{c (a+b x)} (d+e x)^4 \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^4\,c^4\,d^4\,{\ln \left (F\right )}^4+4\,b^4\,c^4\,d^3\,e\,x\,{\ln \left (F\right )}^4+6\,b^4\,c^4\,d^2\,e^2\,x^2\,{\ln \left (F\right )}^4+4\,b^4\,c^4\,d\,e^3\,x^3\,{\ln \left (F\right )}^4+b^4\,c^4\,e^4\,x^4\,{\ln \left (F\right )}^4-4\,b^3\,c^3\,d^3\,e\,{\ln \left (F\right )}^3-12\,b^3\,c^3\,d^2\,e^2\,x\,{\ln \left (F\right )}^3-12\,b^3\,c^3\,d\,e^3\,x^2\,{\ln \left (F\right )}^3-4\,b^3\,c^3\,e^4\,x^3\,{\ln \left (F\right )}^3+12\,b^2\,c^2\,d^2\,e^2\,{\ln \left (F\right )}^2+24\,b^2\,c^2\,d\,e^3\,x\,{\ln \left (F\right )}^2+12\,b^2\,c^2\,e^4\,x^2\,{\ln \left (F\right )}^2-24\,b\,c\,d\,e^3\,\ln \left (F\right )-24\,b\,c\,e^4\,x\,\ln \left (F\right )+24\,e^4\right )}{b^5\,c^5\,{\ln \left (F\right )}^5}
\]
[In]
int(F^(c*(a + b*x))*(d + e*x)^4,x)
[Out]
(F^(a*c + b*c*x)*(24*e^4 + b^4*c^4*d^4*log(F)^4 - 24*b*c*e^4*x*log(F) - 4*b^3*c^3*d^3*e*log(F)^3 + 12*b^2*c^2*
d^2*e^2*log(F)^2 + 12*b^2*c^2*e^4*x^2*log(F)^2 - 4*b^3*c^3*e^4*x^3*log(F)^3 + b^4*c^4*e^4*x^4*log(F)^4 - 24*b*
c*d*e^3*log(F) + 6*b^4*c^4*d^2*e^2*x^2*log(F)^4 + 24*b^2*c^2*d*e^3*x*log(F)^2 + 4*b^4*c^4*d^3*e*x*log(F)^4 - 1
2*b^3*c^3*d^2*e^2*x*log(F)^3 - 12*b^3*c^3*d*e^3*x^2*log(F)^3 + 4*b^4*c^4*d*e^3*x^3*log(F)^4))/(b^5*c^5*log(F)^
5)