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

Optimal. Leaf size=439 \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{3 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{3 a^3 d}+\frac{d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.31895, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391, 266, 36, 29, 31} \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{3 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}-\frac{d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac{(c+d x)^3}{3 a^3 d}+\frac{d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac{(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [F]  time = 1.30192, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [B]  time = 0.089, size = 1999, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/n^2/g^3/f^3/ln(F)^3/a^3*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)))-
1/n/g^3/f^3/ln(F)^3/a^3*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+1/
n/g/f/ln(F)/a^3*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2+1/n/g^3/f^3/ln(F)^3/a^3*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+1/n/g^3/f^3/ln(F)^3/a^3*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^2/f^2/ln(F)^2/a^3*c*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^3*c*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(
F)*f*g*x-ln(F^(g*(f*x+e))))))-2/n^2/g^2/f^2/ln(F)^2/a^3*c*d*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^
(g*(f*x+e)))))/a)-2/n^2/g^2/f^2/ln(F)^2/a^3*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-1/n/g/f/ln(F)/a^3*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^2/g^2/f^2/ln
(F)^2/a^3*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+3/n^2/g^3/f^3/ln(F)^3/a^3*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)))+3/n^2/g^2/f^2/ln(F)^2/a^3*d^2*ln(a+b*F^(n*g*
f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x-3/n^2/g^3/f^3/ln(F)^3/a^3*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)))+1/2*(2*ln(F)*b*d^2*f*g*n*x^2*(F^(g*(f*x+e)))^n+3*ln(F)*a*d^2*f*g
*n*x^2+4*ln(F)*b*c*d*f*g*n*x*(F^(g*(f*x+e)))^n+6*ln(F)*a*c*d*f*g*n*x+2*ln(F)*b*c^2*f*g*n*(F^(g*(f*x+e)))^n+3*l
n(F)*a*c^2*f*g*n-2*b*d^2*x*(F^(g*(f*x+e)))^n-2*a*d^2*x-2*b*c*d*(F^(g*(f*x+e)))^n-2*a*c*d)/n^2/g^2/f^2/ln(F)^2/
a^2/(a+b*(F^(g*(f*x+e)))^n)^2-2/3/g^3/f^3/ln(F)^3/a^3*d^2*ln(F^(g*(f*x+e)))^3+2/n/g/f/ln(F)/a^3*c*d*ln(F^(n*g*
f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x-2/n/g/f/ln(F)/a^3*c*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-l
n(F^(g*(f*x+e))))))*x+2/n/g^2/f^2/ln(F)^2/a^3*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-2/n/g^2/f^2/ln(F)^2/a^3*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-2/n/g^2/f^2/ln(F)^2/a^3*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-2/n/g^2/f^2/ln(F)^2/a^3*c*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)))-2/n/g^2/f^2/ln(F)^2/a^3*c*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
)))+2/n/g^2/f^2/ln(F)^2/a^3*c*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)))+
2/n^3/g^3/f^3/ln(F)^3/a^3*d^2*polylog(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^3*c^2*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*c^2*ln(a+b*F^(n*g*f*x)*exp
(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))-1/n^3/g^3/f^3/ln(F)^3/a^3*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(
F^(g*(f*x+e))))))+1/n^3/g^3/f^3/ln(F)^3/a^3*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+3/n^3/
g^3/f^3/ln(F)^3/a^3*d^2*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-3/2/n/g^3/f^3/ln(F
)^3/a^3*d^2*ln(F^(g*(f*x+e)))^2+1/g^2/f^2/ln(F)^2/a^3*c*d*ln(F^(g*(f*x+e)))^2+1/g^2/f^2/ln(F)^2/a^3*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*} \frac{1}{2} \, c^{2}{\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^{2} f g n x^{2} \log \left (F\right ) - 2 \, a c d + 2 \,{\left ({\left (F^{e g}\right )}^{n} b d^{2} f g n x^{2} \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b c d +{\left (2 \,{\left (F^{e g}\right )}^{n} b c d f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b d^{2}\right )} x\right )}{\left (F^{f g x}\right )}^{n} + 2 \,{\left (3 \, a c d f g n \log \left (F\right ) - a 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{d^{2} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 3 \, c d f g n \log \left (F\right ) + d^{2} +{\left (2 \, c d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, d^{2} f g n \log \left (F\right )\right )} x}{{\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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*((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^2*f*g*n*x^2*log(F) - 2*a*c*d + 2*((F^(e*g))^n*b*d^2*f*g*n*x^2*log(F) - (F^(e*g))^n*b*c
*d + (2*(F^(e*g))^n*b*c*d*f*g*n*log(F) - (F^(e*g))^n*b*d^2)*x)*(F^(f*g*x))^n + 2*(3*a*c*d*f*g*n*log(F) - a*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((d^2*f^2*g^2*n^2*x^2*log(F)^2 - 3*c*d*f*g*n*log(F) + d
^2 + (2*c*d*f^2*g^2*n^2*log(F)^2 - 3*d^2*f*g*n*log(F))*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 = 2.09948, size = 3272, normalized size = 7.45 \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)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a*c**2*f*g*n*log(F) + 6*a*c*d*f*g*n*x*log(F) - 2*a*c*d + 3*a*d**2*f*g*n*x**2*log(F) - 2*a*d**2*x + (2*b*c**
2*f*g*n*log(F) + 4*b*c*d*f*g*n*x*log(F) - 2*b*c*d + 2*b*d**2*f*g*n*x**2*log(F) - 2*b*d**2*x)*(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(d**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log
(F))), x) + Integral(c**2*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) + Integra
l(-3*c*d*f*g*n*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-3*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(d**2*f**2*g**2*n**2*x**2*log(F)**2/(a + b*exp(e*g*n*l
og(F))*exp(f*g*n*x*log(F))), x) + Integral(2*c*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))/(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 )}^{2}}{{\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)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="giac")

[Out]

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