\(\int \frac {(c+d x)^3}{(a+b (F^{g (e+f x)})^n)^2} \, dx\) [52]

Optimal result

Integrand size = 25, antiderivative size = 388 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {3 d (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^4}{4 a^2 d}+\frac {(c+d x)^3}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[In]

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

[Out]

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

Rule 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 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 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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx}{a} \\ & = \frac {(c+d x)^4}{4 a^2 d}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2}-\frac {(3 d) \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a f g n \log (F)} \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {(3 d) \int (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f g n \log (F)}+\frac {(3 b d) \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a^2 f g n \log (F)} \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^2 g^2 n^2 \log ^2(F)} \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {6 d^2 (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (6 d^3\right ) \int \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a^2 f^3 g^3 n^3 \log ^3(F)} \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {6 d^2 (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (6 d^3\right ) \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^2 f^4 g^4 n^3 \log ^4(F)}-\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a^2 f^4 g^4 n^3 \log ^4(F)} \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {6 d^2 (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {6 d^3 \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \text {Li}_4\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3318\) vs. \(2(386)=772\).

Time = 0.46 (sec) , antiderivative size = 3319, normalized size of antiderivative = 8.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(3319\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1390 vs. \(2 (384) = 768\).

Time = 0.31 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.58 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=c^{3} {\left (\frac {f g n x + e g n}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + e g n} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + e g n} b + a\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x}{F^{f g n x} F^{e g n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} - \frac {3 \, c^{2} d x}{a^{2} f g n \log \left (F\right )} + \frac {3 \, c^{2} d \log \left (F^{f g n x} F^{e g n} b + a\right )}{a^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {3 \, {\left (c^{2} d f g n \log \left (F\right ) - 2 \, c 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^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (f^{3} g^{3} n^{3} x^{3} \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right )^{3} + 3 \, f^{2} g^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right ) \log \left (F\right )^{2} - 6 \, f g n x \log \left (F\right ) {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{f g n x} F^{e g n} b}{a})\right )} d^{3}}{a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} - \frac {3 \, {\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 )} {\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )}}{a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {d^{3} f^{4} g^{4} n^{4} x^{4} \log \left (F\right )^{4} + 4 \, {\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 6 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 2 \, c d^{2} f g n \log \left (F\right )\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{4 \, a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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