3.1.59 \(\int \frac {(c+d x)^2}{(a+b (F^{g (e+f x)})^n)^3} \, dx\) [59]

Optimal. Leaf size=439 \[ \frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.94, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438, 272, 36, 29, 31} \begin {gather*} -\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 {(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 {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 {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}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\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} \end {gather*}

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 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]

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 272

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 2215

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 2216

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2222

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 2317

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 2320

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 2438

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 2611

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 6724

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)^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 \text {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 \text {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 ) \text {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 \text {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 ) \text {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 \text {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 ) \text {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*}

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Mathematica [F]
time = 1.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(433)=866\).
time = 0.08, size = 1887, normalized size = 4.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1887\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.40, size = 711, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, c^{2} {\left (\frac {2 \, F^{f g n x + g n e} b + 3 \, a}{{\left (2 \, F^{f g n x + g n e} a^{3} b + F^{2 \, f g n x + 2 \, g n e} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {2 \, {\left (f g n x + g n e\right )}}{a^{3} f g n} - \frac {2 \, \log \left (F^{f g n x + g n e} b + a\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 (F^{g n e} b d^{2} f g n x^{2} \log \left (F\right ) - F^{g n e} b c d + {\left (2 \, F^{g n e} b c d f g n \log \left (F\right ) - F^{g n e} b d^{2}\right )} x\right )} F^{f g n x} + 2 \, {\left (3 \, a c d f g n \log \left (F\right ) - a d^{2}\right )} x}{2 \, {\left (2 \, F^{f g n x} F^{g n e} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x} F^{2 \, g n e} 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 )}} - \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} x}{a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} d^{2}}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} {\left (f g n x \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right )\right )}}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} \log \left (F^{f g n x} F^{g n e} b + a\right )}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{6 \, a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1652 vs. \(2 (440) = 880\).
time = 0.41, size = 1652, normalized size = 3.76 \begin {gather*} \frac {2 \, {\left (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 + 3 \, a^{2} c^{2} f^{2} g^{3} n^{3} e - 3 \, a^{2} c d f g^{3} n^{3} e^{2} + a^{2} d^{2} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{3} + 9 \, {\left (a^{2} c^{2} f^{2} g^{2} n^{2} - 2 \, a^{2} c d f g^{2} n^{2} e + a^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\left (2 \, {\left (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 + 3 \, b^{2} c^{2} f^{2} g^{3} n^{3} e - 3 \, b^{2} c d f g^{3} n^{3} e^{2} + b^{2} d^{2} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{3} - 9 \, {\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + 2 \, b^{2} c d f g^{2} n^{2} e - b^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + 6 \, {\left (b^{2} d^{2} f g n x + b^{2} d^{2} g n e\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 2 \, {\left (2 \, {\left (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 + 3 \, a b c^{2} f^{2} g^{3} n^{3} e - 3 \, a b c d f g^{3} n^{3} e^{2} + a b d^{2} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{3} - 3 \, {\left (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 - a b c^{2} f^{2} g^{2} n^{2} + 6 \, a b c d f g^{2} n^{2} e - 3 \, a b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + 3 \, {\left (a b d^{2} f g n x - a b c d f g n + 2 \, a b d^{2} g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} + 6 \, {\left (3 \, a^{2} d^{2} + {\left (3 \, b^{2} d^{2} - 2 \, {\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 2 \, {\left (3 \, a b d^{2} - 2 \, {\left (a b d^{2} f g n x + a b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 2 \, {\left (a^{2} d^{2} f g n x + a^{2} c d f g n\right )} \log \left (F\right )\right )} {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right ) - 6 \, {\left (a^{2} d^{2} + {\left (a^{2} c^{2} f^{2} g^{2} n^{2} - 2 \, a^{2} c d f g^{2} n^{2} e + a^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\left (b^{2} d^{2} + {\left (b^{2} c^{2} f^{2} g^{2} n^{2} - 2 \, b^{2} c d f g^{2} n^{2} e + b^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 3 \, {\left (b^{2} c d f g n - b^{2} d^{2} g n e\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 2 \, {\left (a b d^{2} + {\left (a b c^{2} f^{2} g^{2} n^{2} - 2 \, a b c d f g^{2} n^{2} e + a b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 3 \, {\left (a b c d f g n - a b d^{2} g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 3 \, {\left (a^{2} c d f g n - a^{2} d^{2} g n e\right )} \log \left (F\right )\right )} \log \left (F^{f g n x + g n e} b + a\right ) - 6 \, {\left (a^{2} c d f g n - a^{2} d^{2} g n e\right )} \log \left (F\right ) - 6 \, {\left ({\left (a^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} c d f^{2} g^{2} n^{2} x + 2 \, a^{2} c d f g^{2} n^{2} e - a^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} + {\left ({\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + 2 \, b^{2} c d f g^{2} n^{2} e - b^{2} d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 3 \, {\left (b^{2} d^{2} f g n x + b^{2} d^{2} g n e\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 2 \, {\left ({\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + 2 \, a b c d f g^{2} n^{2} e - a b d^{2} g^{2} n^{2} e^{2}\right )} \log \left (F\right )^{2} - 3 \, {\left (a b d^{2} f g n x + a b d^{2} g n e\right )} \log \left (F\right )\right )} F^{f g n x + g n e} - 3 \, {\left (a^{2} d^{2} f g n x + a^{2} d^{2} g n e\right )} \log \left (F\right )\right )} \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right ) + 12 \, {\left (2 \, F^{f g n x + g n e} a b d^{2} + F^{2 \, f g n x + 2 \, g n e} b^{2} d^{2} + a^{2} d^{2}\right )} {\rm polylog}\left (3, -\frac {F^{f g n x + g n e} b}{a}\right )}{6 \, {\left (2 \, F^{f g n x + g n e} a^{4} b f^{3} g^{3} n^{3} \log \left (F\right )^{3} + F^{2 \, f g n x + 2 \, g n e} a^{3} b^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3} + a^{5} f^{3} g^{3} n^{3} \log \left (F\right )^{3}\right )}} \end {gather*}

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*(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 + 3*a^2*c^2*f^2*g^3*n^3*
e - 3*a^2*c*d*f*g^3*n^3*e^2 + a^2*d^2*g^3*n^3*e^3)*log(F)^3 + 9*(a^2*c^2*f^2*g^2*n^2 - 2*a^2*c*d*f*g^2*n^2*e +
a^2*d^2*g^2*n^2*e^2)*log(F)^2 + (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 + 3*b^2*c^2*f^2*g^3*n^3*e - 3*b^2*c*d*f*g^3*n^3*e^2 + b^2*d^2*g^3*n^3*e^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 + 2*b^2*c*d*f*g^2*n^2*e - b^2*d^2*g^2*n^2*e^2)*log(F)^2 + 6*(b^2*d^2*f*g*n*x
+ b^2*d^2*g*n*e)*log(F))*F^(2*f*g*n*x + 2*g*n*e) + 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 + 3*a*b*c^2*f^2*g^3*n^3*e - 3*a*b*c*d*f*g^3*n^3*e^2 + a*b*d^2*g^3*n^3*e^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 - a*b*c^2*f^2*g^2*n^2 + 6*a*b*c*d*f*g^2*n^2*e - 3*a*b
*d^2*g^2*n^2*e^2)*log(F)^2 + 3*(a*b*d^2*f*g*n*x - a*b*c*d*f*g*n + 2*a*b*d^2*g*n*e)*log(F))*F^(f*g*n*x + g*n*e)
+ 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*g*n*e) + 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 + g*n*e) - 2*(a^2*d^2*f*g*n*x + a^2*c*d*f*g*n)*lo
g(F))*dilog(-(F^(f*g*n*x + g*n*e)*b + a)/a + 1) - 6*(a^2*d^2 + (a^2*c^2*f^2*g^2*n^2 - 2*a^2*c*d*f*g^2*n^2*e +
a^2*d^2*g^2*n^2*e^2)*log(F)^2 + (b^2*d^2 + (b^2*c^2*f^2*g^2*n^2 - 2*b^2*c*d*f*g^2*n^2*e + b^2*d^2*g^2*n^2*e^2)
*log(F)^2 - 3*(b^2*c*d*f*g*n - b^2*d^2*g*n*e)*log(F))*F^(2*f*g*n*x + 2*g*n*e) + 2*(a*b*d^2 + (a*b*c^2*f^2*g^2*
n^2 - 2*a*b*c*d*f*g^2*n^2*e + a*b*d^2*g^2*n^2*e^2)*log(F)^2 - 3*(a*b*c*d*f*g*n - a*b*d^2*g*n*e)*log(F))*F^(f*g
*n*x + g*n*e) - 3*(a^2*c*d*f*g*n - a^2*d^2*g*n*e)*log(F))*log(F^(f*g*n*x + g*n*e)*b + a) - 6*(a^2*c*d*f*g*n -
a^2*d^2*g*n*e)*log(F) - 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 + 2*a^2*c*d*f*g^2*n^2*e - a^2*d^
2*g^2*n^2*e^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 + 2*b^2*c*d*f*g^2*n^2*e - b^2*d^
2*g^2*n^2*e^2)*log(F)^2 - 3*(b^2*d^2*f*g*n*x + b^2*d^2*g*n*e)*log(F))*F^(2*f*g*n*x + 2*g*n*e) + 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 + 2*a*b*c*d*f*g^2*n^2*e - a*b*d^2*g^2*n^2*e^2)*log(F)^2 - 3*(a*b*d^2*f
*g*n*x + a*b*d^2*g*n*e)*log(F))*F^(f*g*n*x + g*n*e) - 3*(a^2*d^2*f*g*n*x + a^2*d^2*g*n*e)*log(F))*log((F^(f*g*
n*x + g*n*e)*b + a)/a) + 12*(2*F^(f*g*n*x + g*n*e)*a*b*d^2 + F^(2*f*g*n*x + 2*g*n*e)*b^2*d^2 + a^2*d^2)*polylo
g(3, -F^(f*g*n*x + g*n*e)*b/a))/(2*F^(f*g*n*x + g*n*e)*a^4*b*f^3*g^3*n^3*log(F)^3 + F^(2*f*g*n*x + 2*g*n*e)*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \left (- \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 )}}}\right )\, dx + \int \left (- \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 )}}}\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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