3.45 \(\int \frac{(a+b (F^{g (e+f x)})^n)^3}{(c+d x)^3} \, dx\)

Optimal. Leaf size=447 \[ \frac{3 a^2 b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{a^3}{2 d (c+d x)^2}+\frac{6 a b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{3 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}+\frac{9 b^3 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n}}{2 d^2 (c+d x)}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.694877, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2183, 2177, 2182, 2178} \[ \frac{3 a^2 b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n}{2 d^2 (c+d x)}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{a^3}{2 d (c+d x)^2}+\frac{6 a b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^3}-\frac{3 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}+\frac{9 b^3 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{2 d^3}-\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n}}{2 d^2 (c+d x)}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^3,x]

[Out]

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

Rule 2183

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2177

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

Rule 2182

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*F^(g*(e +
f*x)))^n/F^(g*n*(e + f*x)), Int[(c + d*x)^m*F^(g*n*(e + f*x)), x], x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3}{(c+d x)^3} \, dx &=\int \left (\frac{a^3}{(c+d x)^3}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{(c+d x)^3}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^3}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{(c+d x)^3}\right ) \, dx\\ &=-\frac{a^3}{2 d (c+d x)^2}+\left (3 a^2 b\right ) \int \frac{\left (F^{e g+f g x}\right )^n}{(c+d x)^3} \, dx+\left (3 a b^2\right ) \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^3} \, dx+b^3 \int \frac{\left (F^{e g+f g x}\right )^{3 n}}{(c+d x)^3} \, dx\\ &=-\frac{a^3}{2 d (c+d x)^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2}+\frac{\left (3 a^2 b f g n \log (F)\right ) \int \frac{\left (F^{e g+f g x}\right )^n}{(c+d x)^2} \, dx}{2 d}+\frac{\left (3 a b^2 f g n \log (F)\right ) \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^2} \, dx}{d}+\frac{\left (3 b^3 f g n \log (F)\right ) \int \frac{\left (F^{e g+f g x}\right )^{3 n}}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac{a^3}{2 d (c+d x)^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2}-\frac{3 a^2 b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}-\frac{3 a b^2 f \left (F^{e g+f g x}\right )^{2 n} g n \log (F)}{d^2 (c+d x)}-\frac{3 b^3 f \left (F^{e g+f g x}\right )^{3 n} g n \log (F)}{2 d^2 (c+d x)}+\frac{\left (3 a^2 b f^2 g^2 n^2 \log ^2(F)\right ) \int \frac{\left (F^{e g+f g x}\right )^n}{c+d x} \, dx}{2 d^2}+\frac{\left (6 a b^2 f^2 g^2 n^2 \log ^2(F)\right ) \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{c+d x} \, dx}{d^2}+\frac{\left (9 b^3 f^2 g^2 n^2 \log ^2(F)\right ) \int \frac{\left (F^{e g+f g x}\right )^{3 n}}{c+d x} \, dx}{2 d^2}\\ &=-\frac{a^3}{2 d (c+d x)^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2}-\frac{3 a^2 b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}-\frac{3 a b^2 f \left (F^{e g+f g x}\right )^{2 n} g n \log (F)}{d^2 (c+d x)}-\frac{3 b^3 f \left (F^{e g+f g x}\right )^{3 n} g n \log (F)}{2 d^2 (c+d x)}+\frac{\left (3 a^2 b f^2 F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n g^2 n^2 \log ^2(F)\right ) \int \frac{F^{n (e g+f g x)}}{c+d x} \, dx}{2 d^2}+\frac{\left (6 a b^2 f^2 F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n} g^2 n^2 \log ^2(F)\right ) \int \frac{F^{2 n (e g+f g x)}}{c+d x} \, dx}{d^2}+\frac{\left (9 b^3 f^2 F^{-3 n (e g+f g x)} \left (F^{e g+f g x}\right )^{3 n} g^2 n^2 \log ^2(F)\right ) \int \frac{F^{3 n (e g+f g x)}}{c+d x} \, dx}{2 d^2}\\ &=-\frac{a^3}{2 d (c+d x)^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{2 d (c+d x)^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{2 d (c+d x)^2}-\frac{3 a^2 b f \left (F^{e g+f g x}\right )^n g n \log (F)}{2 d^2 (c+d x)}-\frac{3 a b^2 f \left (F^{e g+f g x}\right )^{2 n} g n \log (F)}{d^2 (c+d x)}-\frac{3 b^3 f \left (F^{e g+f g x}\right )^{3 n} g n \log (F)}{2 d^2 (c+d x)}+\frac{3 a^2 b f^2 F^{\left (e-\frac{c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n g^2 n^2 \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right ) \log ^2(F)}{2 d^3}+\frac{6 a b^2 f^2 F^{2 \left (e-\frac{c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} g^2 n^2 \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right ) \log ^2(F)}{d^3}+\frac{9 b^3 f^2 F^{3 \left (e-\frac{c f}{d}\right ) g n-3 g n (e+f x)} \left (F^{e g+f g x}\right )^{3 n} g^2 n^2 \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right ) \log ^2(F)}{2 d^3}\\ \end{align*}

Mathematica [A]  time = 1.02305, size = 325, normalized size = 0.73 \[ -\frac{-3 a^2 b f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )+3 a^2 b d \left (F^{g (e+f x)}\right )^n (f g n \log (F) (c+d x)+d)+a^3 d^2-12 a b^2 f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )+3 a b^2 d \left (F^{g (e+f x)}\right )^{2 n} (2 f g n \log (F) (c+d x)+d)-9 b^3 f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^{3 n} F^{-\frac{3 f g n (c+d x)}{d}} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )+b^3 d \left (F^{g (e+f x)}\right )^{3 n} (3 f g n \log (F) (c+d x)+d)}{2 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^3,x]

[Out]

-(a^3*d^2 - (3*a^2*b*f^2*(F^(g*(e + f*x)))^n*g^2*n^2*(c + d*x)^2*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log
[F]^2)/F^((f*g*n*(c + d*x))/d) - (12*a*b^2*f^2*(F^(g*(e + f*x)))^(2*n)*g^2*n^2*(c + d*x)^2*ExpIntegralEi[(2*f*
g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/F^((2*f*g*n*(c + d*x))/d) - (9*b^3*f^2*(F^(g*(e + f*x)))^(3*n)*g^2*n^2*(c +
d*x)^2*ExpIntegralEi[(3*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/F^((3*f*g*n*(c + d*x))/d) + 3*a^2*b*d*(F^(g*(e +
f*x)))^n*(d + f*g*n*(c + d*x)*Log[F]) + 3*a*b^2*d*(F^(g*(e + f*x)))^(2*n)*(d + 2*f*g*n*(c + d*x)*Log[F]) + b^
3*d*(F^(g*(e + f*x)))^(3*n)*(d + 3*f*g*n*(c + d*x)*Log[F]))/(2*d^3*(c + d*x)^2)

________________________________________________________________________________________

Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^3,x)

[Out]

int((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^3,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (F^{e g}\right )}^{3 \, n} b^{3} \int \frac{{\left (F^{f g x}\right )}^{3 \, n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + 3 \,{\left (F^{e g}\right )}^{2 \, n} a b^{2} \int \frac{{\left (F^{f g x}\right )}^{2 \, n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + 3 \,{\left (F^{e g}\right )}^{n} a^{2} b \int \frac{{\left (F^{f g x}\right )}^{n}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} - \frac{a^{3}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

(F^(e*g))^(3*n)*b^3*integrate((F^(f*g*x))^(3*n)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) + 3*(F^(e*g))^(2
*n)*a*b^2*integrate((F^(f*g*x))^(2*n)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) + 3*(F^(e*g))^n*a^2*b*inte
grate((F^(f*g*x))^n/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x) - 1/2*a^3/(d^3*x^2 + 2*c*d^2*x + c^2*d)

________________________________________________________________________________________

Fricas [A]  time = 1.58188, size = 1018, normalized size = 2.28 \begin{align*} -\frac{a^{3} d^{2} - 9 \,{\left (b^{3} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d f^{2} g^{2} n^{2} x + b^{3} c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac{3 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{3 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )^{2} - 12 \,{\left (a b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d f^{2} g^{2} n^{2} x + a b^{2} c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )^{2} - 3 \,{\left (a^{2} b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d f^{2} g^{2} n^{2} x + a^{2} b c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )^{2} +{\left (b^{3} d^{2} + 3 \,{\left (b^{3} d^{2} f g n x + b^{3} c d f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, e g n} + 3 \,{\left (a b^{2} d^{2} + 2 \,{\left (a b^{2} d^{2} f g n x + a b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} + 3 \,{\left (a^{2} b d^{2} +{\left (a^{2} b d^{2} f g n x + a^{2} b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(a^3*d^2 - 9*(b^3*d^2*f^2*g^2*n^2*x^2 + 2*b^3*c*d*f^2*g^2*n^2*x + b^3*c^2*f^2*g^2*n^2)*F^(3*(d*e - c*f)*g
*n/d)*Ei(3*(d*f*g*n*x + c*f*g*n)*log(F)/d)*log(F)^2 - 12*(a*b^2*d^2*f^2*g^2*n^2*x^2 + 2*a*b^2*c*d*f^2*g^2*n^2*
x + a*b^2*c^2*f^2*g^2*n^2)*F^(2*(d*e - c*f)*g*n/d)*Ei(2*(d*f*g*n*x + c*f*g*n)*log(F)/d)*log(F)^2 - 3*(a^2*b*d^
2*f^2*g^2*n^2*x^2 + 2*a^2*b*c*d*f^2*g^2*n^2*x + a^2*b*c^2*f^2*g^2*n^2)*F^((d*e - c*f)*g*n/d)*Ei((d*f*g*n*x + c
*f*g*n)*log(F)/d)*log(F)^2 + (b^3*d^2 + 3*(b^3*d^2*f*g*n*x + b^3*c*d*f*g*n)*log(F))*F^(3*f*g*n*x + 3*e*g*n) +
3*(a*b^2*d^2 + 2*(a*b^2*d^2*f*g*n*x + a*b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 3*(a^2*b*d^2 + (a^2*b
*d^2*f*g*n*x + a^2*b*c*d*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F**(g*(f*x+e)))**n)**3/(d*x+c)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

integrate(((F^((f*x + e)*g))^n*b + a)^3/(d*x + c)^3, x)