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

Optimal. Leaf size=594 \[ \frac{9 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d^3 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{9 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{3 d^2 (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{(c+d x)^3}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^4}{4 a^3 d}+\frac{(c+d x)^3}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

[Out]

(c + d*x)^4/(4*a^3*d) + (3*d*(c + d*x)^2)/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (3*d*(c + d*x)^2)/(2*a^2*f^2*(a + b*(
F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*x)^3)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^3/(2*a*f*(a + b*(F^(g
*(e + f*x)))^n)^2*g*n*Log[F]) + (c + d*x)^3/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (3*d^2*(c + d*x)*
Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^3*g^3*n^3*Log[F]^3) + (9*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))
^n)/a])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (
3*d^3*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (9*d^2*(c + d*x)*PolyLog[2, -((b*
(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a
)])/(a^3*f^2*g^2*n^2*Log[F]^2) - (9*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) +
(6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((
b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4)

________________________________________________________________________________________

Rubi [A]  time = 1.90083, antiderivative size = 594, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191, 2279, 2391} \[ \frac{9 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d^3 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{9 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{3 d^2 (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{(c+d x)^3}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^4}{4 a^3 d}+\frac{(c+d x)^3}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^4/(4*a^3*d) + (3*d*(c + d*x)^2)/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (3*d*(c + d*x)^2)/(2*a^2*f^2*(a + b*(
F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*x)^3)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^3/(2*a*f*(a + b*(F^(g
*(e + f*x)))^n)^2*g*n*Log[F]) + (c + d*x)^3/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (3*d^2*(c + d*x)*
Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^3*g^3*n^3*Log[F]^3) + (9*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))
^n)/a])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (
3*d^3*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (9*d^2*(c + d*x)*PolyLog[2, -((b*
(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a
)])/(a^3*f^2*g^2*n^2*Log[F]^2) - (9*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) +
(6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((
b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4)

Rule 2185

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2184

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

Rule 2190

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

Rule 2531

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

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2191

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p +
1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx &=\frac{\int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx}{a}\\ &=\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{\int \frac{(c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a^2}-\frac{(3 d) \int \frac{(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{2 a f g n \log (F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{b \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^3}-\frac{(3 d) \int \frac{(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{2 a^2 f g n \log (F)}-\frac{(3 d) \int \frac{(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)}+\frac{(3 b d) \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{2 a^2 f g n \log (F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}-\frac{3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{(c+d x)^3 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac{\left (3 d^2\right ) \int \frac{c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac{(3 d) \int (c+d x)^2 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f g n \log (F)}+\frac{(3 b d) \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{2 a^3 f g n \log (F)}+\frac{(3 b d) \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^3 f g n \log (F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac{9 d (c+d x)^2 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac{3 d (c+d x)^2 \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{\left (3 d^2\right ) \int (c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{\left (6 d^2\right ) \int (c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{\left (6 d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{\left (3 b d^2\right ) \int \frac{\left (F^{g (e+f x)}\right )^n (c+d x)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^3 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{3 d^2 (c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac{9 d^2 (c+d x) \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{6 d^2 (c+d x) \text{Li}_3\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{\left (3 d^3\right ) \int \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{\left (3 d^3\right ) \int \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{\left (6 d^3\right ) \int \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{\left (6 d^3\right ) \int \text{Li}_3\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^3 f^3 g^3 n^3 \log ^3(F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{3 d^2 (c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac{9 d^2 (c+d x) \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{6 d^2 (c+d x) \text{Li}_3\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a^3 f^4 g^4 n^3 \log ^4(F)}-\frac{\left (6 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a^3 f^4 g^4 n^3 \log ^4(F)}-\frac{\left (6 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a^3 f^4 g^4 n^3 \log ^4(F)}\\ &=\frac{(c+d x)^4}{4 a^3 d}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac{(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{3 d^2 (c+d x) \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (1+\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac{3 d^3 \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}+\frac{9 d^2 (c+d x) \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{Li}_2\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{9 d^3 \text{Li}_3\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}+\frac{6 d^2 (c+d x) \text{Li}_3\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{6 d^3 \text{Li}_4\left (-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}\\ \end{align*}

Mathematica [F]  time = 1.72784, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.147, size = 4237, normalized size = 7.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/n/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x-
6/n/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))*
x-6/n/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x-6/
n^2/g^2/f^2/ln(F)^2/a^3*c*d^2*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*x-3/n/g^2/f^
2/ln(F)^2/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x^2+3/n/g^3/f^3/ln
(F)^3/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2*x-3/n/g^2/f^2/ln(F)^
2/a^3*d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))*x^2+3/n/g^3/f^3/ln(F
)^3/a^3*d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))^2*x+3/n/g/f/ln(F)/
a^3*c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2+3/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(F^(n*g*
f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2-3/n/g/f/ln(F)/a^3*c*d^2*ln(a+b*F^(n*g*f*x)*e
xp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2-3/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g
*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2+3/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g
*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))^2+3/n/g/f/ln(F)/a^3*c^2*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^
(g*(f*x+e))))))*x-3/n/g/f/ln(F)/a^3*c^2*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+3/n/g^
2/f^2/ln(F)^2/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x^2-3/n/g^
3/f^3/ln(F)^3/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2*x-9/n^2/
g^2/f^2/ln(F)^2/a^3*c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+9/n^2/g^3/f^3/ln(F)^3/a^3*
c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))+1/2*(2*ln(F)*b*d^3*f*g*n*x^3*(
F^(g*(f*x+e)))^n+3*ln(F)*a*d^3*f*g*n*x^3+6*ln(F)*b*c*d^2*f*g*n*x^2*(F^(g*(f*x+e)))^n+9*ln(F)*a*c*d^2*f*g*n*x^2
+6*ln(F)*b*c^2*d*f*g*n*x*(F^(g*(f*x+e)))^n+9*ln(F)*a*c^2*d*f*g*n*x+2*ln(F)*b*c^3*f*g*n*(F^(g*(f*x+e)))^n+3*ln(
F)*a*c^3*f*g*n-3*b*d^3*x^2*(F^(g*(f*x+e)))^n-3*a*d^3*x^2-6*b*c*d^2*x*(F^(g*(f*x+e)))^n-6*a*c*d^2*x-3*b*c^2*d*(
F^(g*(f*x+e)))^n-3*a*c^2*d)/n^2/g^2/f^2/ln(F)^2/a^2/(a+b*(F^(g*(f*x+e)))^n)^2-3/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*
ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+6/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*polylog(3,-b*F^(n*
g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)+9/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*polylog(2,-b*F^(n*g*f*x)*exp
(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)+6/n^3/g^3/f^3/ln(F)^3/a^3*d^3*polylog(3,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f
*g*x-ln(F^(g*(f*x+e)))))/a)*x-1/n/g/f/ln(F)/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))
)*x^3+1/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e))
)^3+9/2/n^2/g^2/f^2/ln(F)^2/a^3*c^2*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))-3/n^2/g^2/f^
2/ln(F)^2/a^3*c^2*d*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-3/n^2/g^2/f^2/ln(F)^2/
a^3*d^3*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*x^2-9/2/n^2/g^2/f^2/ln(F)^2/a^3*c^
2*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+1/n/g/f/ln(F)/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)
*f*g*x-ln(F^(g*(f*x+e))))))*x^3-1/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)
)))))*ln(F^(g*(f*x+e)))^3-3/n^3/g^3/f^3/ln(F)^3/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)
)))))*x+3/n^3/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x
+e)))-1/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e
)))^3-3/n^3/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x
+e)))+3/4/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(g*(f*x+e)))^4+3/n^3/g^3/f^3/ln(F)^3/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(
F)*f*g*x-ln(F^(g*(f*x+e))))))*x-3/n^3/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+
e))))))*ln(F^(g*(f*x+e)))-9/2/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(F^(g*(f*x+e)))^2+9/n^3/g^3/f^3/ln(F)^3/a^3*d^3*po
lylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*x-9/2/n/g^3/f^3/ln(F)^3/a^3*d^3*ln(F^(g*(f*x
+e)))^2*x+9/2/n^2/g^2/f^2/ln(F)^2/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2+9/2/
n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2-9/
2/n^2/g^2/f^2/ln(F)^2/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2-9/2/n^2/g^4/f^4/ln(F
)^4/a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2-9/2/n^2/g^4/f^4/ln(F)^
4/a^3*d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))^2+3/n^3/g^3/f^3/ln(F
)^3/a^3*c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+3/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(F^(g*(f*x
+e)))^2*x+3/2/g^2/f^2/ln(F)^2/a^3*c^2*d*ln(F^(g*(f*x+e)))^2+3/2/g^2/f^2/ln(F)^2/a^3*d^3*ln(F^(g*(f*x+e)))^2*x^
2-2/g^3/f^3/ln(F)^3/a^3*d^3*ln(F^(g*(f*x+e)))^3*x-2/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(F^(g*(f*x+e)))^3+3/2/n^2/g^4/
f^4/ln(F)^4/a^3*d^3*ln(F^(g*(f*x+e)))^2+3/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(g*(f*x+e)))^3+1/n/g/f/ln(F)/a^3*c^3*
ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))-6/n^4/g^4/f^4/ln(F)^4/a^3*d^3*polylog(4,-b*F^(n*g*f*x)
*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-3/n^4/g^4/f^4/ln(F)^4/a^3*d^3*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(
F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-1/n/g/f/ln(F)/a^3*c^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))
)))-9/n^4/g^4/f^4/ln(F)^4/a^3*d^3*polylog(3,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)+9/n^2/g^
3/f^3/ln(F)^3/a^3*c*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))+9/n^2/
g^2/f^2/ln(F)^2/a^3*c*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x-9/n^2/g^3/f^3/ln(F)^3/
a^3*c*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))-9/n^2/g^3/f^3/ln(F)^3/
a^3*d^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x+9/n^2/g^3/f^3/ln(F)^3/
a^3*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x+9/n^2/g^3/f^3/ln(F)^3/a^3*
d^3*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))*x-3/n/g^2/f^2/ln(F)^2/a^3*
c^2*d*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))-3/n/g^2/f^2/ln(F)^2/a^3*
c^2*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))+3/n/g^2/f^2/ln(F)^2/a^3*c^2*d*
ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, c^{3}{\left (\frac{2 \,{\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{{\left (2 \,{\left (F^{f g x + e g}\right )}^{n} a^{3} b n +{\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \left (F\right )} + \frac{2 \, \log \left (F^{f g x + e g}\right )}{a^{3} f g \log \left (F\right )} - \frac{2 \, \log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac{3 \, a d^{3} f g n x^{3} \log \left (F\right ) - 3 \, a c^{2} d + 3 \,{\left (3 \, a c d^{2} f g n \log \left (F\right ) - a d^{3}\right )} x^{2} +{\left (2 \,{\left (F^{e g}\right )}^{n} b d^{3} f g n x^{3} \log \left (F\right ) - 3 \,{\left (F^{e g}\right )}^{n} b c^{2} d + 3 \,{\left (2 \,{\left (F^{e g}\right )}^{n} b c d^{2} f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b d^{3}\right )} x^{2} + 6 \,{\left ({\left (F^{e g}\right )}^{n} b c^{2} d f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b c d^{2}\right )} x\right )}{\left (F^{f g x}\right )}^{n} + 3 \,{\left (3 \, a c^{2} d f g n \log \left (F\right ) - 2 \, a c d^{2}\right )} x}{2 \,{\left (2 \,{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} +{\left (F^{f g x}\right )}^{2 \, n}{\left (F^{e g}\right )}^{2 \, n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} + \int \frac{2 \, d^{3} f^{2} g^{2} n^{2} x^{3} \log \left (F\right )^{2} - 9 \, c^{2} d f g n \log \left (F\right ) + 6 \, c d^{2} + 3 \,{\left (2 \, c d^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, d^{3} f g n \log \left (F\right )\right )} x^{2} + 6 \,{\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\right )} x}{2 \,{\left ({\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^3*((2*(F^(f*g*x + e*g))^n*b + 3*a)/((2*(F^(f*g*x + e*g))^n*a^3*b*n + (F^(f*g*x + e*g))^(2*n)*a^2*b^2*n +
a^4*n)*f*g*log(F)) + 2*log(F^(f*g*x + e*g))/(a^3*f*g*log(F)) - 2*log(((F^(f*g*x + e*g))^n*b + a)/b)/(a^3*f*g*
n*log(F))) + 1/2*(3*a*d^3*f*g*n*x^3*log(F) - 3*a*c^2*d + 3*(3*a*c*d^2*f*g*n*log(F) - a*d^3)*x^2 + (2*(F^(e*g))
^n*b*d^3*f*g*n*x^3*log(F) - 3*(F^(e*g))^n*b*c^2*d + 3*(2*(F^(e*g))^n*b*c*d^2*f*g*n*log(F) - (F^(e*g))^n*b*d^3)
*x^2 + 6*((F^(e*g))^n*b*c^2*d*f*g*n*log(F) - (F^(e*g))^n*b*c*d^2)*x)*(F^(f*g*x))^n + 3*(3*a*c^2*d*f*g*n*log(F)
- 2*a*c*d^2)*x)/(2*(F^(f*g*x))^n*(F^(e*g))^n*a^3*b*f^2*g^2*n^2*log(F)^2 + (F^(f*g*x))^(2*n)*(F^(e*g))^(2*n)*a
^2*b^2*f^2*g^2*n^2*log(F)^2 + a^4*f^2*g^2*n^2*log(F)^2) + integrate(1/2*(2*d^3*f^2*g^2*n^2*x^3*log(F)^2 - 9*c^
2*d*f*g*n*log(F) + 6*c*d^2 + 3*(2*c*d^2*f^2*g^2*n^2*log(F)^2 - 3*d^3*f*g*n*log(F))*x^2 + 6*(c^2*d*f^2*g^2*n^2*
log(F)^2 - 3*c*d^2*f*g*n*log(F) + d^3)*x)/((F^(f*g*x))^n*(F^(e*g))^n*a^2*b*f^2*g^2*n^2*log(F)^2 + a^3*f^2*g^2*
n^2*log(F)^2), x)

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Fricas [C]  time = 2.41398, size = 5646, normalized size = 9.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*g^3*n^3*log(F)^3 + 6*(a^2*d^3*e^2
- 2*a^2*c*d^2*e*f + a^2*c^2*d*f^2)*g^2*n^2*log(F)^2 - (a^2*d^3*f^4*g^4*n^4*x^4 + 4*a^2*c*d^2*f^4*g^4*n^4*x^3 +
6*a^2*c^2*d*f^4*g^4*n^4*x^2 + 4*a^2*c^3*f^4*g^4*n^4*x - (a^2*d^3*e^4 - 4*a^2*c*d^2*e^3*f + 6*a^2*c^2*d*e^2*f^
2 - 4*a^2*c^3*e*f^3)*g^4*n^4)*log(F)^4 - ((b^2*d^3*f^4*g^4*n^4*x^4 + 4*b^2*c*d^2*f^4*g^4*n^4*x^3 + 6*b^2*c^2*d
*f^4*g^4*n^4*x^2 + 4*b^2*c^3*f^4*g^4*n^4*x - (b^2*d^3*e^4 - 4*b^2*c*d^2*e^3*f + 6*b^2*c^2*d*e^2*f^2 - 4*b^2*c^
3*e*f^3)*g^4*n^4)*log(F)^4 - 6*(b^2*d^3*f^3*g^3*n^3*x^3 + 3*b^2*c*d^2*f^3*g^3*n^3*x^2 + 3*b^2*c^2*d*f^3*g^3*n^
3*x + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 + 6*(b^2*d^3*f^2*g^2*n^2*x^2 + 2
*b^2*c*d^2*f^2*g^2*n^2*x - (b^2*d^3*e^2 - 2*b^2*c*d^2*e*f)*g^2*n^2)*log(F)^2)*F^(2*f*g*n*x + 2*e*g*n) - 2*((a*
b*d^3*f^4*g^4*n^4*x^4 + 4*a*b*c*d^2*f^4*g^4*n^4*x^3 + 6*a*b*c^2*d*f^4*g^4*n^4*x^2 + 4*a*b*c^3*f^4*g^4*n^4*x -
(a*b*d^3*e^4 - 4*a*b*c*d^2*e^3*f + 6*a*b*c^2*d*e^2*f^2 - 4*a*b*c^3*e*f^3)*g^4*n^4)*log(F)^4 - 2*(2*a*b*d^3*f^3
*g^3*n^3*x^3 + 6*a*b*c*d^2*f^3*g^3*n^3*x^2 + 6*a*b*c^2*d*f^3*g^3*n^3*x + (3*a*b*d^3*e^3 - 9*a*b*c*d^2*e^2*f +
9*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*g^3*n^3)*log(F)^3 + 3*(a*b*d^3*f^2*g^2*n^2*x^2 + 2*a*b*c*d^2*f^2*g^2*n^2*x -
(2*a*b*d^3*e^2 - 4*a*b*c*d^2*e*f + a*b*c^2*d*f^2)*g^2*n^2)*log(F)^2)*F^(f*g*n*x + e*g*n) + 12*(a^2*d^3 + (a^2*
d^3*f^2*g^2*n^2*x^2 + 2*a^2*c*d^2*f^2*g^2*n^2*x + a^2*c^2*d*f^2*g^2*n^2)*log(F)^2 + (b^2*d^3 + (b^2*d^3*f^2*g^
2*n^2*x^2 + 2*b^2*c*d^2*f^2*g^2*n^2*x + b^2*c^2*d*f^2*g^2*n^2)*log(F)^2 - 3*(b^2*d^3*f*g*n*x + b^2*c*d^2*f*g*n
)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(a*b*d^3 + (a*b*d^3*f^2*g^2*n^2*x^2 + 2*a*b*c*d^2*f^2*g^2*n^2*x + a*b*c^
2*d*f^2*g^2*n^2)*log(F)^2 - 3*(a*b*d^3*f*g*n*x + a*b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + e*g*n) - 3*(a^2*d^3*f*g
*n*x + a^2*c*d^2*f*g*n)*log(F))*dilog(-(F^(f*g*n*x + e*g*n)*b + a)/a + 1) - 2*(2*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^
2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*g^3*n^3*log(F)^3 + 9*(a^2*d^3*e^2 - 2*a^2*c*d^2*e*f + a^2*c^2*d*f^2)*g^
2*n^2*log(F)^2 + 6*(a^2*d^3*e - a^2*c*d^2*f)*g*n*log(F) + (2*(b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*
f^2 - b^2*c^3*f^3)*g^3*n^3*log(F)^3 + 9*(b^2*d^3*e^2 - 2*b^2*c*d^2*e*f + b^2*c^2*d*f^2)*g^2*n^2*log(F)^2 + 6*(
b^2*d^3*e - b^2*c*d^2*f)*g*n*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c
^2*d*e*f^2 - a*b*c^3*f^3)*g^3*n^3*log(F)^3 + 9*(a*b*d^3*e^2 - 2*a*b*c*d^2*e*f + a*b*c^2*d*f^2)*g^2*n^2*log(F)^
2 + 6*(a*b*d^3*e - a*b*c*d^2*f)*g*n*log(F))*F^(f*g*n*x + e*g*n))*log(F^(f*g*n*x + e*g*n)*b + a) + 2*(2*(a^2*d^
3*f^3*g^3*n^3*x^3 + 3*a^2*c*d^2*f^3*g^3*n^3*x^2 + 3*a^2*c^2*d*f^3*g^3*n^3*x + (a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f
+ 3*a^2*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 9*(a^2*d^3*f^2*g^2*n^2*x^2 + 2*a^2*c*d^2*f^2*g^2*n^2*x - (a^2*d^3*e^
2 - 2*a^2*c*d^2*e*f)*g^2*n^2)*log(F)^2 + (2*(b^2*d^3*f^3*g^3*n^3*x^3 + 3*b^2*c*d^2*f^3*g^3*n^3*x^2 + 3*b^2*c^2
*d*f^3*g^3*n^3*x + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 9*(b^2*d^3*f^2*g^
2*n^2*x^2 + 2*b^2*c*d^2*f^2*g^2*n^2*x - (b^2*d^3*e^2 - 2*b^2*c*d^2*e*f)*g^2*n^2)*log(F)^2 + 6*(b^2*d^3*f*g*n*x
+ b^2*d^3*e*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(a*b*d^3*f^3*g^3*n^3*x^3 + 3*a*b*c*d^2*f^3*g^3*n^3*x^
2 + 3*a*b*c^2*d*f^3*g^3*n^3*x + (a*b*d^3*e^3 - 3*a*b*c*d^2*e^2*f + 3*a*b*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 9*(a
*b*d^3*f^2*g^2*n^2*x^2 + 2*a*b*c*d^2*f^2*g^2*n^2*x - (a*b*d^3*e^2 - 2*a*b*c*d^2*e*f)*g^2*n^2)*log(F)^2 + 6*(a*
b*d^3*f*g*n*x + a*b*d^3*e*g*n)*log(F))*F^(f*g*n*x + e*g*n) + 6*(a^2*d^3*f*g*n*x + a^2*d^3*e*g*n)*log(F))*log((
F^(f*g*n*x + e*g*n)*b + a)/a) + 24*(2*F^(f*g*n*x + e*g*n)*a*b*d^3 + F^(2*f*g*n*x + 2*e*g*n)*b^2*d^3 + a^2*d^3)
*polylog(4, -F^(f*g*n*x + e*g*n)*b/a) + 12*(3*a^2*d^3 + (3*b^2*d^3 - 2*(b^2*d^3*f*g*n*x + b^2*c*d^2*f*g*n)*log
(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(3*a*b*d^3 - 2*(a*b*d^3*f*g*n*x + a*b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + e*g*n
) - 2*(a^2*d^3*f*g*n*x + a^2*c*d^2*f*g*n)*log(F))*polylog(3, -F^(f*g*n*x + e*g*n)*b/a))/(2*F^(f*g*n*x + e*g*n)
*a^4*b*f^4*g^4*n^4*log(F)^4 + F^(2*f*g*n*x + 2*e*g*n)*a^3*b^2*f^4*g^4*n^4*log(F)^4 + a^5*f^4*g^4*n^4*log(F)^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a*c**3*f*g*n*log(F) + 9*a*c**2*d*f*g*n*x*log(F) - 3*a*c**2*d + 9*a*c*d**2*f*g*n*x**2*log(F) - 6*a*c*d**2*x
+ 3*a*d**3*f*g*n*x**3*log(F) - 3*a*d**3*x**2 + (2*b*c**3*f*g*n*log(F) + 6*b*c**2*d*f*g*n*x*log(F) - 3*b*c**2*d
+ 6*b*c*d**2*f*g*n*x**2*log(F) - 6*b*c*d**2*x + 2*b*d**3*f*g*n*x**3*log(F) - 3*b*d**3*x**2)*(F**(g*(e + f*x))
)**n)/(2*a**4*f**2*g**2*n**2*log(F)**2 + 4*a**3*b*f**2*g**2*n**2*(F**(g*(e + f*x)))**n*log(F)**2 + 2*a**2*b**2
*f**2*g**2*n**2*(F**(g*(e + f*x)))**(2*n)*log(F)**2) + (Integral(6*c*d**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x
*log(F))), x) + Integral(6*d**3*x/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(2*c**3*f**2*g**
2*n**2*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-9*c**2*d*f*g*n*log(F)/(a + b*ex
p(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-9*d**3*f*g*n*x**2*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g
*n*x*log(F))), x) + Integral(2*d**3*f**2*g**2*n**2*x**3*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))
), x) + Integral(-18*c*d**2*f*g*n*x*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(6*c*d*
*2*f**2*g**2*n**2*x**2*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(6*c**2*d*f**2*g*
*2*n**2*x*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x))/(2*a**2*f**2*g**2*n**2*log(F)**2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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