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

Optimal. Leaf size=192 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^4}{4 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.32587, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2184, 2190, 2531, 6609, 2282, 6589} \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac{(c+d x)^4}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.106588, size = 166, normalized size = 0.86 \[ \frac{\frac{3 d \left (f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \text{PolyLog}\left (2,-\frac{a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d \left (f g n \log (F) (c+d x) \text{PolyLog}\left (3,-\frac{a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+d \text{PolyLog}\left (4,-\frac{a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )\right )\right )}{f^3 g^3 n^3 \log ^3(F)}-(c+d x)^3 \log \left (\frac{a \left (F^{g (e+f x)}\right )^{-n}}{b}+1\right )}{a f g n \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.094, size = 3227, normalized size = 16.8 \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),x)

[Out]

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

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

[Out]

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

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Fricas [C]  time = 1.6537, size = 888, normalized size = 4.62 \begin{align*} \frac{4 \,{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} g^{3} n^{3} \log \left (F^{f g n x + e g n} b + a\right ) \log \left (F\right )^{3} +{\left (d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 4 \,{\left (d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, c^{2} d f^{3} g^{3} n^{3} x +{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} g^{3} n^{3}\right )} \log \left (F\right )^{3} \log \left (\frac{F^{f g n x + e g n} b + a}{a}\right ) - 12 \,{\left (d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, c d^{2} f^{2} g^{2} n^{2} x + c^{2} d f^{2} g^{2} n^{2}\right )}{\rm Li}_2\left (-\frac{F^{f g n x + e g n} b + a}{a} + 1\right ) \log \left (F\right )^{2} - 24 \, d^{3}{\rm polylog}\left (4, -\frac{F^{f g n x + e g n} b}{a}\right ) + 24 \,{\left (d^{3} f g n x + c d^{2} f g n\right )} \log \left (F\right ){\rm polylog}\left (3, -\frac{F^{f g n x + e g n} b}{a}\right )}{4 \, a f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \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),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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

[Out]

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