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

Optimal result

Integrand size = 25, antiderivative size = 439 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \]

[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] (verified)

Time = 0.87 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438, 272, 36, 29, 31} \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {3 d (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {(c+d x)^2}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac {(c+d x)^2}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

[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*} \text {integral}& = \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*}

Mathematica [F]

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

[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]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(433)=866\).

Time = 0.43 (sec) , antiderivative size = 1887, normalized size of antiderivative = 4.30

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (431) = 862\).

Time = 0.30 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.46 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Too large to display} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {1}{2} \, c^{2} {\left (\frac {2 \, F^{f g n x + e g n} b + 3 \, a}{{\left (2 \, F^{f g n x + e g n} a^{3} b + F^{2 \, f g n x + 2 \, e g n} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {2 \, {\left (f g n x + e g n\right )}}{a^{3} f g n} - \frac {2 \, \log \left (F^{f g n x + e g n} 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^{e g n} b d^{2} f g n x^{2} \log \left (F\right ) - F^{e g n} b c d + {\left (2 \, F^{e g n} b c d f g n \log \left (F\right ) - F^{e g n} 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^{e g n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x} F^{2 \, e g 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 )}} - \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^{e g n} 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^{e g n} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} 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^{e g n} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} 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^{e g n} 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}} \]

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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \]

[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)