3.2 \(\int F^{c (a+b x)} (d+e x)^4 \, dx\)

Optimal. Leaf size=141 \[ \frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^3 (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 (d+e x)^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {4 e (d+e x)^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)} \]

[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)

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Rubi [A]  time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2176, 2194} \[ \frac {12 e^2 (d+e x)^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {24 e^3 (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {4 e (d+e x)^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}+\frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[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 2176

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] &&  !$UseGamma === True

Rule 2194

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*} \int F^{c (a+b x)} (d+e x)^4 \, dx &=\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*}

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Mathematica [A]  time = 0.13, size = 100, normalized size = 0.71 \[ \frac {F^{c (a+b x)} \left (b^4 c^4 \log ^4(F) (d+e x)^4-4 b^3 c^3 e \log ^3(F) (d+e x)^3+12 b^2 c^2 e^2 \log ^2(F) (d+e x)^2-24 b c e^3 \log (F) (d+e x)+24 e^4\right )}{b^5 c^5 \log ^5(F)} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.65, size = 227, normalized size = 1.61 \[ \frac {{\left ({\left (b^{4} c^{4} e^{4} x^{4} + 4 \, b^{4} c^{4} d e^{3} x^{3} + 6 \, b^{4} c^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} c^{4} d^{3} e x + b^{4} c^{4} d^{4}\right )} \log \relax (F)^{4} + 24 \, e^{4} - 4 \, {\left (b^{3} c^{3} e^{4} x^{3} + 3 \, b^{3} c^{3} d e^{3} x^{2} + 3 \, b^{3} c^{3} d^{2} e^{2} x + b^{3} c^{3} d^{3} e\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} c^{2} e^{4} x^{2} + 2 \, b^{2} c^{2} d e^{3} x + b^{2} c^{2} d^{2} e^{2}\right )} \log \relax (F)^{2} - 24 \, {\left (b c e^{4} x + b c d e^{3}\right )} \log \relax (F)\right )} F^{b c x + a c}}{b^{5} c^{5} \log \relax (F)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [C]  time = 1.67, size = 7867, normalized size = 55.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^4,x, algorithm="giac")

[Out]

-((4*(pi^3*b^4*c^4*x^4*log(abs(F))*sgn(F) - pi*b^4*c^4*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*x^4*log(abs(F))
+ pi*b^4*c^4*x^4*log(abs(F))^3 - pi^3*b^3*c^3*x^3*sgn(F) + 3*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*c
^3*x^3 - 3*pi*b^3*c^3*x^3*log(abs(F))^2 - 6*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*c^2*x^2*log(abs(F)) +
6*pi*b*c*x*sgn(F) - 6*pi*b*c*x)*(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*lo
g(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*x^4*sgn(F) - 6*pi^2*b^4*c^4*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*x^4 + 6*pi^2*b^4*c^4*x^4*lo
g(abs(F))^2 - 2*b^4*c^4*x^4*log(abs(F))^4 + 12*pi^2*b^3*c^3*x^3*log(abs(F))*sgn(F) - 12*pi^2*b^3*c^3*x^3*log(a
bs(F)) + 8*b^3*c^3*x^3*log(abs(F))^3 - 12*pi^2*b^2*c^2*x^2*sgn(F) + 12*pi^2*b^2*c^2*x^2 - 24*b^2*c^2*x^2*log(a
bs(F))^2 + 48*b*c*x*log(abs(F)) - 48)*(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))*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*x^4*sgn(F) - 6*pi^2*b^4*c
^4*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*x^4 + 6*pi^2*b^4*c^4*x^4*log(abs(F))^2 - 2*b^4*c^4*x^4*log(abs(F))^
4 + 12*pi^2*b^3*c^3*x^3*log(abs(F))*sgn(F) - 12*pi^2*b^3*c^3*x^3*log(abs(F)) + 8*b^3*c^3*x^3*log(abs(F))^3 - 1
2*pi^2*b^2*c^2*x^2*sgn(F) + 12*pi^2*b^2*c^2*x^2 - 24*b^2*c^2*x^2*log(abs(F))^2 + 48*b*c*x*log(abs(F)) - 48)*(p
i^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) + 4*(pi^3*b^4*c^4*x^4*log(abs(F))*sg
n(F) - pi*b^4*c^4*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*x^4*log(abs(F)) + pi*b^4*c^4*x^4*log(abs(F))^3 - pi^
3*b^3*c^3*x^3*sgn(F) + 3*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*x^3 - 3*pi*b^3*c^3*x^3*log(abs(F))
^2 - 6*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*c^2*x^2*log(abs(F)) + 6*pi*b*c*x*sgn(F) - 6*pi*b*c*x)*(5*p
i^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*sgn(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)) + 4) + 1/2*I*((-16*I*pi^4*b^4*c^4*x^4*sgn(F)
+ 64*pi^3*b^4*c^4*x^4*log(abs(F))*sgn(F) + 96*I*pi^2*b^4*c^4*x^4*log(abs(F))^2*sgn(F) - 64*pi*b^4*c^4*x^4*log
(abs(F))^3*sgn(F) + 16*I*pi^4*b^4*c^4*x^4 - 64*pi^3*b^4*c^4*x^4*log(abs(F)) - 96*I*pi^2*b^4*c^4*x^4*log(abs(F)
)^2 + 64*pi*b^4*c^4*x^4*log(abs(F))^3 + 32*I*b^4*c^4*x^4*log(abs(F))^4 - 64*pi^3*b^3*c^3*x^3*sgn(F) - 192*I*pi
^2*b^3*c^3*x^3*log(abs(F))*sgn(F) + 192*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) + 64*pi^3*b^3*c^3*x^3 + 192*I*pi^2
*b^3*c^3*x^3*log(abs(F)) - 192*pi*b^3*c^3*x^3*log(abs(F))^2 - 128*I*b^3*c^3*x^3*log(abs(F))^3 + 192*I*pi^2*b^2
*c^2*x^2*sgn(F) - 384*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 192*I*pi^2*b^2*c^2*x^2 + 384*pi*b^2*c^2*x^2*log(abs(
F)) + 384*I*b^2*c^2*x^2*log(abs(F))^2 + 384*pi*b*c*x*sgn(F) - 384*pi*b*c*x - 768*I*b*c*x*log(abs(F)) + 768*I)*
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*sg
n(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
) - (-16*I*pi^4*b^4*c^4*x^4*sgn(F) - 64*pi^3*b^4*c^4*x^4*log(abs(F))*sgn(F) + 96*I*pi^2*b^4*c^4*x^4*log(abs(F)
)^2*sgn(F) + 64*pi*b^4*c^4*x^4*log(abs(F))^3*sgn(F) + 16*I*pi^4*b^4*c^4*x^4 + 64*pi^3*b^4*c^4*x^4*log(abs(F))
- 96*I*pi^2*b^4*c^4*x^4*log(abs(F))^2 - 64*pi*b^4*c^4*x^4*log(abs(F))^3 + 32*I*b^4*c^4*x^4*log(abs(F))^4 + 64*
pi^3*b^3*c^3*x^3*sgn(F) - 192*I*pi^2*b^3*c^3*x^3*log(abs(F))*sgn(F) - 192*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F)
- 64*pi^3*b^3*c^3*x^3 + 192*I*pi^2*b^3*c^3*x^3*log(abs(F)) + 192*pi*b^3*c^3*x^3*log(abs(F))^2 - 128*I*b^3*c^3*
x^3*log(abs(F))^3 + 192*I*pi^2*b^2*c^2*x^2*sgn(F) + 384*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 192*I*pi^2*b^2*c^2
*x^2 - 384*pi*b^2*c^2*x^2*log(abs(F)) + 384*I*b^2*c^2*x^2*log(abs(F))^2 - 384*pi*b*c*x*sgn(F) + 384*pi*b*c*x -
768*I*b*c*x*log(abs(F)) + 768*I)*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)) + 4) - 4*(((3*pi^2*b^3*c^3*d*x^3
*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*d*x^3*log(abs(F)) + 2*b^3*c^3*d*x^3*log(abs(F))^3 - 3*pi^2*b^2*c^2*d*x^2*
sgn(F) + 3*pi^2*b^2*c^2*d*x^2 - 6*b^2*c^2*d*x^2*log(abs(F))^2 + 12*b*c*d*x*log(abs(F)) - 12*d)*(pi^4*b^4*c^4*s
gn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(
F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))
^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3
*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*c^3*d*x^3*sgn(F) - 3*pi*b^3*c^3*d*x^3*log(ab
s(F))^2*sgn(F) - pi^3*b^3*c^3*d*x^3 + 3*pi*b^3*c^3*d*x^3*log(abs(F))^2 + 6*pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F)
- 6*pi*b^2*c^2*d*x^2*log(abs(F)) - 6*pi*b*c*d*x*sgn(F) + 6*pi*b*c*d*x)*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*
b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)/((pi^4*b^4*c^4*sgn(F) - 6*
pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 +
16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4
*log(abs(F))^3)^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^3*b^3*c^3
*d*x^3*sgn(F) - 3*pi*b^3*c^3*d*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*d*x^3 + 3*pi*b^3*c^3*d*x^3*log(abs(F))^
2 + 6*pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*d*x^2*log(abs(F)) - 6*pi*b*c*d*x*sgn(F) + 6*pi*b*c*d*
x)*(pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 -
2*b^4*c^4*log(abs(F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b
^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F
))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2) + 4*(3*pi^2*b^3*c^3*d*x^3*log(abs(F))*sg
n(F) - 3*pi^2*b^3*c^3*d*x^3*log(abs(F)) + 2*b^3*c^3*d*x^3*log(abs(F))^3 - 3*pi^2*b^2*c^2*d*x^2*sgn(F) + 3*pi^2
*b^2*c^2*d*x^2 - 6*b^2*c^2*d*x^2*log(abs(F))^2 + 12*b*c*d*x*log(abs(F)) - 12*d)*(pi^3*b^4*c^4*log(abs(F))*sgn(
F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)/((pi^4*b^4*c^4*sgn
(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F)
)^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi
*b^4*c^4*log(abs(F))^3)^2))*sin(-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))*e^(b*c*
x*log(abs(F)) + a*c*log(abs(F)) + 3) - 1/2*I*((32*pi^3*b^3*c^3*d*x^3*sgn(F) + 96*I*pi^2*b^3*c^3*d*x^3*log(abs(
F))*sgn(F) - 96*pi*b^3*c^3*d*x^3*log(abs(F))^2*sgn(F) - 32*pi^3*b^3*c^3*d*x^3 - 96*I*pi^2*b^3*c^3*d*x^3*log(ab
s(F)) + 96*pi*b^3*c^3*d*x^3*log(abs(F))^2 + 64*I*b^3*c^3*d*x^3*log(abs(F))^3 - 96*I*pi^2*b^2*c^2*d*x^2*sgn(F)
+ 192*pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F) + 96*I*pi^2*b^2*c^2*d*x^2 - 192*pi*b^2*c^2*d*x^2*log(abs(F)) - 192*I
*b^2*c^2*d*x^2*log(abs(F))^2 - 192*pi*b*c*d*x*sgn(F) + 192*pi*b*c*d*x + 384*I*b*c*d*x*log(abs(F)) - 384*I*d)*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)/(8*pi^4*b^4*c^4*sgn(F) + 32*I*p
i^3*b^4*c^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - 32*I*pi*b^4*c^4*log(abs(F))^3*sgn(F) -
8*pi^4*b^4*c^4 - 32*I*pi^3*b^4*c^4*log(abs(F)) + 48*pi^2*b^4*c^4*log(abs(F))^2 + 32*I*pi*b^4*c^4*log(abs(F))^
3 - 16*b^4*c^4*log(abs(F))^4) + (32*pi^3*b^3*c^3*d*x^3*sgn(F) - 96*I*pi^2*b^3*c^3*d*x^3*log(abs(F))*sgn(F) - 9
6*pi*b^3*c^3*d*x^3*log(abs(F))^2*sgn(F) - 32*pi^3*b^3*c^3*d*x^3 + 96*I*pi^2*b^3*c^3*d*x^3*log(abs(F)) + 96*pi*
b^3*c^3*d*x^3*log(abs(F))^2 - 64*I*b^3*c^3*d*x^3*log(abs(F))^3 + 96*I*pi^2*b^2*c^2*d*x^2*sgn(F) + 192*pi*b^2*c
^2*d*x^2*log(abs(F))*sgn(F) - 96*I*pi^2*b^2*c^2*d*x^2 - 192*pi*b^2*c^2*d*x^2*log(abs(F)) + 192*I*b^2*c^2*d*x^2
*log(abs(F))^2 - 192*pi*b*c*d*x*sgn(F) + 192*pi*b*c*d*x - 384*I*b*c*d*x*log(abs(F)) + 384*I*d)*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)/(8*pi^4*b^4*c^4*sgn(F) - 32*I*pi^3*b^4*c^4*l
og(abs(F))*sgn(F) - 48*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) + 32*I*pi*b^4*c^4*log(abs(F))^3*sgn(F) - 8*pi^4*b^4*c
^4 + 32*I*pi^3*b^4*c^4*log(abs(F)) + 48*pi^2*b^4*c^4*log(abs(F))^2 - 32*I*pi*b^4*c^4*log(abs(F))^3 - 16*b^4*c^
4*log(abs(F))^4))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 3) - 6*((2*(pi*b^2*c^2*d^2*x^2*log(abs(F))*sgn(F) -
pi*b^2*c^2*d^2*x^2*log(abs(F)) - pi*b*c*d^2*x*sgn(F) + pi*b*c*d^2*x)*(pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(
abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))
^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3
*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)^2) - (pi^2*b^2*c^2*d^2*x^2*sgn(F) - pi^2*b^2*c^2*d^2*x^2 + 2*b^2*c^2*d
^2*x^2*log(abs(F))^2 - 4*b*c*d^2*x*log(abs(F)) + 4*d^2)*(3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*lo
g(abs(F)) + 2*b^3*c^3*log(abs(F))^3)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3
+ 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*
log(abs(F))^3)^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^2*b^2*c^2*
d^2*x^2*sgn(F) - pi^2*b^2*c^2*d^2*x^2 + 2*b^2*c^2*d^2*x^2*log(abs(F))^2 - 4*b*c*d^2*x*log(abs(F)) + 4*d^2)*(pi
^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)/((pi^3*b^3*
c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^
3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)^2) + 2*(pi*b^2*c^2*d^2*x^2*log(ab
s(F))*sgn(F) - pi*b^2*c^2*d^2*x^2*log(abs(F)) - pi*b*c*d^2*x*sgn(F) + pi*b*c*d^2*x)*(3*pi^2*b^3*c^3*log(abs(F)
)*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(
F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*
c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)^2))*sin(-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))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 2) + 1/2*I*((24*I*pi^2*b^2*c^2*d^2*x^2*sgn(F) - 48*pi*b^2
*c^2*d^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b^2*c^2*d^2*x^2 + 48*pi*b^2*c^2*d^2*x^2*log(abs(F)) + 48*I*b^2*c^2
*d^2*x^2*log(abs(F))^2 + 48*pi*b*c*d^2*x*sgn(F) - 48*pi*b*c*d^2*x - 96*I*b*c*d^2*x*log(abs(F)) + 96*I*d^2)*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)/(-4*I*pi^3*b^3*c^3*sgn(F) + 12*pi
^2*b^3*c^3*log(abs(F))*sgn(F) + 12*I*pi*b^3*c^3*log(abs(F))^2*sgn(F) + 4*I*pi^3*b^3*c^3 - 12*pi^2*b^3*c^3*log(
abs(F)) - 12*I*pi*b^3*c^3*log(abs(F))^2 + 8*b^3*c^3*log(abs(F))^3) - (24*I*pi^2*b^2*c^2*d^2*x^2*sgn(F) + 48*pi
*b^2*c^2*d^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b^2*c^2*d^2*x^2 - 48*pi*b^2*c^2*d^2*x^2*log(abs(F)) + 48*I*b^2
*c^2*d^2*x^2*log(abs(F))^2 - 48*pi*b*c*d^2*x*sgn(F) + 48*pi*b*c*d^2*x - 96*I*b*c*d^2*x*log(abs(F)) + 96*I*d^2)
*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)/(4*I*pi^3*b^3*c^3*sgn(F) + 1
2*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 12*I*pi*b^3*c^3*log(abs(F))^2*sgn(F) - 4*I*pi^3*b^3*c^3 - 12*pi^2*b^3*c^3*
log(abs(F)) + 12*I*pi*b^3*c^3*log(abs(F))^2 + 8*b^3*c^3*log(abs(F))^3))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))
+ 2) + 4*(2*((b*c*d^3*x*log(abs(F)) - d^3)*(pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)/((p
i^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2
*log(abs(F)))^2) + (pi*b*c*d^3*x*sgn(F) - pi*b*c*d^3*x)*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)
))/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b
^2*c^2*log(abs(F)))^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*b*c*d
^3*x*sgn(F) - pi*b*c*d^3*x)*(pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)/((pi^2*b^2*c^2*sgn(
F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2)
- 4*(b*c*d^3*x*log(abs(F)) - d^3)*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))/((pi^2*b^2*c^2*sgn
(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2
))*sin(-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))*e^(b*c*x*log(abs(F)) + a*c*log(a
bs(F)) + 1) - 1/2*I*((8*pi*b*c*d^3*x*sgn(F) - 8*pi*b*c*d^3*x - 16*I*b*c*d^3*x*log(abs(F)) + 16*I*d^3)*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)/(2*pi^2*b^2*c^2*sgn(F) + 4*I*pi*b^2*c^
2*log(abs(F))*sgn(F) - 2*pi^2*b^2*c^2 - 4*I*pi*b^2*c^2*log(abs(F)) + 4*b^2*c^2*log(abs(F))^2) + (8*pi*b*c*d^3*
x*sgn(F) - 8*pi*b*c*d^3*x + 16*I*b*c*d^3*x*log(abs(F)) - 16*I*d^3)*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)/(2*pi^2*b^2*c^2*sgn(F) - 4*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 2*pi^2*b^2*
c^2 + 4*I*pi*b^2*c^2*log(abs(F)) + 4*b^2*c^2*log(abs(F))^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 1) + 2*(
2*b*c*d^4*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)*log(abs(F))/(4*b^2*c^2*log
(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*d^4*sin(-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)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F
)) + a*c*log(abs(F))) - 1/2*I*(-2*I*d^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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))) + 2*I*d^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)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)
) + a*c*log(abs(F)))

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maple [A]  time = 0.01, size = 260, normalized size = 1.84 \[ \frac {\left (b^{4} c^{4} e^{4} x^{4} \ln \relax (F )^{4}+4 b^{4} c^{4} d \,e^{3} x^{3} \ln \relax (F )^{4}+6 b^{4} c^{4} d^{2} e^{2} x^{2} \ln \relax (F )^{4}+4 b^{4} c^{4} d^{3} e x \ln \relax (F )^{4}+b^{4} c^{4} d^{4} \ln \relax (F )^{4}-4 b^{3} c^{3} e^{4} x^{3} \ln \relax (F )^{3}-12 b^{3} c^{3} d \,e^{3} x^{2} \ln \relax (F )^{3}-12 b^{3} c^{3} d^{2} e^{2} x \ln \relax (F )^{3}-4 b^{3} c^{3} d^{3} e \ln \relax (F )^{3}+12 b^{2} c^{2} e^{4} x^{2} \ln \relax (F )^{2}+24 b^{2} c^{2} d \,e^{3} x \ln \relax (F )^{2}+12 b^{2} c^{2} d^{2} e^{2} \ln \relax (F )^{2}-24 b c \,e^{4} x \ln \relax (F )-24 b c d \,e^{3} \ln \relax (F )+24 e^{4}\right ) F^{\left (b x +a \right ) c}}{b^{5} c^{5} \ln \relax (F )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^((b*x+a)*c)*(e*x+d)^4,x)

[Out]

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

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maxima [B]  time = 0.50, size = 309, normalized size = 2.19 \[ \frac {F^{b c x + a c} d^{4}}{b c \log \relax (F)} + \frac {4 \, {\left (F^{a c} b c x \log \relax (F) - F^{a c}\right )} F^{b c x} d^{3} e}{b^{2} c^{2} \log \relax (F)^{2}} + \frac {6 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} - 2 \, F^{a c} b c x \log \relax (F) + 2 \, F^{a c}\right )} F^{b c x} d^{2} e^{2}}{b^{3} c^{3} \log \relax (F)^{3}} + \frac {4 \, {\left (F^{a c} b^{3} c^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{a c} b c x \log \relax (F) - 6 \, F^{a c}\right )} F^{b c x} d e^{3}}{b^{4} c^{4} \log \relax (F)^{4}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \relax (F)^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \relax (F)^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} - 24 \, F^{a c} b c x \log \relax (F) + 24 \, F^{a c}\right )} F^{b c x} e^{4}}{b^{5} c^{5} \log \relax (F)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.68, size = 260, normalized size = 1.84 \[ \frac {F^{a\,c+b\,c\,x}\,\left (b^4\,c^4\,d^4\,{\ln \relax (F)}^4+4\,b^4\,c^4\,d^3\,e\,x\,{\ln \relax (F)}^4+6\,b^4\,c^4\,d^2\,e^2\,x^2\,{\ln \relax (F)}^4+4\,b^4\,c^4\,d\,e^3\,x^3\,{\ln \relax (F)}^4+b^4\,c^4\,e^4\,x^4\,{\ln \relax (F)}^4-4\,b^3\,c^3\,d^3\,e\,{\ln \relax (F)}^3-12\,b^3\,c^3\,d^2\,e^2\,x\,{\ln \relax (F)}^3-12\,b^3\,c^3\,d\,e^3\,x^2\,{\ln \relax (F)}^3-4\,b^3\,c^3\,e^4\,x^3\,{\ln \relax (F)}^3+12\,b^2\,c^2\,d^2\,e^2\,{\ln \relax (F)}^2+24\,b^2\,c^2\,d\,e^3\,x\,{\ln \relax (F)}^2+12\,b^2\,c^2\,e^4\,x^2\,{\ln \relax (F)}^2-24\,b\,c\,d\,e^3\,\ln \relax (F)-24\,b\,c\,e^4\,x\,\ln \relax (F)+24\,e^4\right )}{b^5\,c^5\,{\ln \relax (F)}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.27, size = 350, normalized size = 2.48 \[ \begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{4} \log {\relax (F )}^{4} + 4 b^{4} c^{4} d^{3} e x \log {\relax (F )}^{4} + 6 b^{4} c^{4} d^{2} e^{2} x^{2} \log {\relax (F )}^{4} + 4 b^{4} c^{4} d e^{3} x^{3} \log {\relax (F )}^{4} + b^{4} c^{4} e^{4} x^{4} \log {\relax (F )}^{4} - 4 b^{3} c^{3} d^{3} e \log {\relax (F )}^{3} - 12 b^{3} c^{3} d^{2} e^{2} x \log {\relax (F )}^{3} - 12 b^{3} c^{3} d e^{3} x^{2} \log {\relax (F )}^{3} - 4 b^{3} c^{3} e^{4} x^{3} \log {\relax (F )}^{3} + 12 b^{2} c^{2} d^{2} e^{2} \log {\relax (F )}^{2} + 24 b^{2} c^{2} d e^{3} x \log {\relax (F )}^{2} + 12 b^{2} c^{2} e^{4} x^{2} \log {\relax (F )}^{2} - 24 b c d e^{3} \log {\relax (F )} - 24 b c e^{4} x \log {\relax (F )} + 24 e^{4}\right )}{b^{5} c^{5} \log {\relax (F )}^{5}} & \text {for}\: b^{5} c^{5} \log {\relax (F )}^{5} \neq 0 \\d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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