3.52 \(\int \frac{(c+d x)^3}{(a+b (F^{g (e+f x)})^n)^2} \, dx\)

Optimal. Leaf size=388 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac{3 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^3}{a^2 f g n \log (F)}+\frac{(c+d x)^4}{4 a^2 d}+\frac{(c+d x)^3}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.866812, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac{3 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^3}{a^2 f g n \log (F)}+\frac{(c+d x)^4}{4 a^2 d}+\frac{(c+d x)^3}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

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

Rubi steps

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

Mathematica [F]  time = 1.72344, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

Integrate[(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2, x]

________________________________________________________________________________________

Maple [B]  time = 0.138, size = 3519, normalized size = 9.1 \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)^2,x)

[Out]

-6/n/g^2/f^2/ln(F)^2/a^2*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
/g^2/f^2/ln(F)^2/a^2*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^2*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^3/g^3/f^3/ln(F)^3/a^2*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+6/n^3/g^3/f
^3/ln(F)^3/a^2*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)-3/n^2/g^2/f^2/ln(F)^2
/a^2*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+6/n^3/g^3/f^3/ln(F)^3/a^2*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+6/n^3/g^3/f^3/ln(F)^3/a^2*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)-3/n^2/g^2/f^2/ln(F)^2/a^2*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/g^3/f^3/ln(F)^3/a^2*c*d^2*ln(F^(g*(f*x+e)))^2-1/n/g^4/f
^4/ln(F)^4/a^2*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+1/n/g/f/l
n(F)/a^2*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^2*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)))^3-1/n/g/f/ln(F)/a^2*d^3*ln(a+b*F^(n*g*f*x)*ex
p(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^3+1/n/g^4/f^4/ln(F)^4/a^2*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)))^3-3/n/g^3/f^3/ln(F)^3/a^2*d^3*ln(F^(g*(f*x+e)))^2*x-3/n^2/g^2/f^2/ln(F)
^2/a^2*d^3*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2+3/n^2/g^2/f^2/ln(F)^2/a^2*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+3/n^2/g^4/f^4/ln(F)^4/a^2*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)))^2-3/n^2/g^4/f^4/ln(F)^4/a^2*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+1/n/g/f/ln(F)/a*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(a+b*(F
^(g*(f*x+e)))^n)+3/4/g^4/f^4/ln(F)^4/a^2*d^3*ln(F^(g*(f*x+e)))^4-2/g^3/f^3/ln(F)^3/a^2*c*d^2*ln(F^(g*(f*x+e)))
^3+3/2/g^2/f^2/ln(F)^2/a^2*d^3*ln(F^(g*(f*x+e)))^2*x^2-2/g^3/f^3/ln(F)^3/a^2*d^3*ln(F^(g*(f*x+e)))^3*x+3/2/g^2
/f^2/ln(F)^2/a^2*c^2*d*ln(F^(g*(f*x+e)))^2+2/n/g^4/f^4/ln(F)^4/a^2*d^3*ln(F^(g*(f*x+e)))^3-6/n^4/g^4/f^4/ln(F)
^4/a^2*d^3*polylog(4,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-6/n^4/g^4/f^4/ln(F)^4/a^2*d^3*p
olylog(3,-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^2*c^3*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^2*c^3*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e
))))))-3/n/g^2/f^2/ln(F)^2/a^2*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/f/ln(F)/a^2*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^2/f^2/ln(F)^2/a
^2*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)))-6/n^2/g^2/f^2/ln(F)^2/a^2*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+6/n^2/g^3/f^3/ln(F)^3/a^2*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+6/n^2/g^3/f^3/ln(F)^3/a^2*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+6/n^2/g^3/f^3/ln(F)^3/a^2*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)))-3/n/g^2/f^2/ln(F)^2/a^2*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^2*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-6/n^2/g^3/f^3/ln(F)^3/a^2*d^3*l
n(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-3/n/g/f/ln(F)/a^2*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^2*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)))+3/n/g/f/ln(F)/a^2*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^2*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^2*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^2
-3/n/g^3/f^3/ln(F)^3/a^2*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^2/f^2/ln(F)^2/a^2*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^2*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^2*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^2*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-6/n^2/g
^2/f^2/ln(F)^2/a^2*c*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+6/n^2/g^3/f^3/ln(F)^3/a^2*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)))+6/n^2/g^2/f^2/ln(F)^2/a^2*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-6/n^2/g^3/f^3/ln(F)^3/a^2*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)))+3/n/g^3/f^3/ln(F)^3/a^2*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^2/g^4/f^4/ln(F)^4/a^2*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^2/g^2/f^2/ln(F)^2/a^2*c^2*d*ln(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^2*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/g^2/f^2/ln(F)^2/a^2*c*d^2*ln(F^(g*(f*x+e)))^2*x

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

[Out]

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

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Fricas [C]  time = 1.81476, size = 2936, normalized size = 7.57 \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)^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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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 )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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