\(\int \frac {f^x x^3}{(a+b f^{2 x})^2} \, dx\) [50]

Optimal result

Integrand size = 18, antiderivative size = 501 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {3 x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i x \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)} \]

[Out]

1/2*f^x*x^3/a/(a+b*f^(2*x))/ln(f)-3/2*x^2*arctan(f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^2/b^(1/2)+1/2*x^3*arctan(f
^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)/b^(1/2)+3/2*I*x*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^3/b^(1/2)-3/
4*I*x^2*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^2/b^(1/2)-3/2*I*x*polylog(2,I*f^x*b^(1/2)/a^(1/2))/a^(
3/2)/ln(f)^3/b^(1/2)+3/4*I*x^2*polylog(2,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^2/b^(1/2)-3/2*I*polylog(3,-I*f^x
*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)+3/2*I*x*polylog(3,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^3/b^(1/2)+3/
2*I*polylog(3,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)-3/2*I*x*polylog(3,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/
ln(f)^3/b^(1/2)-3/2*I*polylog(4,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)+3/2*I*polylog(4,I*f^x*b^(1/2)/
a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {2281, 205, 211, 2277, 14, 12, 5251, 2611, 2320, 6724, 6744} \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

[In]

Int[(f^x*x^3)/(a + b*f^(2*x))^2,x]

[Out]

(-3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (f^x*x^3)/(2*a*(a + b*f^(2*x))*Log[f]) +
(x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]) + (((3*I)/2)*x*PolyLog[2, ((-I)*Sqrt[b]*f^x)/S
qrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^3) - (((3*I)/4)*x^2*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*
Log[f]^2) - (((3*I)/2)*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^3) + (((3*I)/4)*x^2*Poly
Log[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2) - (((3*I)/2)*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]
])/(a^(3/2)*Sqrt[b]*Log[f]^4) + (((3*I)/2)*x*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^3
) + (((3*I)/2)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^4) - (((3*I)/2)*x*PolyLog[3, (I*Sq
rt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^3) - (((3*I)/2)*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*
Sqrt[b]*Log[f]^4) + (((3*I)/2)*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2277

Int[(F_)^((e_.)*((c_.) + (d_.)*(x_)))*((a_.) + (b_.)*(F_)^(v_))^(p_)*(x_)^(m_.), x_Symbol] :> With[{u = IntHid
e[F^(e*(c + d*x))*(a + b*F^v)^p, x]}, Dist[x^m, u, x] - Dist[m, Int[x^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c,
d, e}, x] && EqQ[v, 2*e*(c + d*x)] && GtQ[m, 0] && ILtQ[p, 0]

Rule 2281

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[d*e*(Log[F]/(g*h*Log[G]))]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - d*e*(f/g))*x^Numerator[m])^p, x], x, G^(h*((f + g*x)/Denominator[m]))], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

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 5251

Int[ArcTan[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] :> Dist[I/2, Int[x^m*Log[1 - I*a - I
*b*f^(c + d*x)], x], x] - Dist[I/2, Int[x^m*Log[1 + I*a + I*b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x
] && IntegerQ[m] && 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 {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-3 \int x^2 \left (\frac {f^x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-3 \int \left (\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 \int \frac {f^x x^2}{a+b f^{2 x}} \, dx}{2 a \log (f)}-\frac {3 \int x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{a \log (f)}-\frac {(3 i) \int x^2 \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{3/2} \sqrt {b} \log (f)}+\frac {(3 i) \int x^2 \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{3/2} \sqrt {b} \log (f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {(3 i) \int x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 \int x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{a^{3/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {(3 i) \int \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {(3 i) \int \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)} \\ & = -\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.87 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {\frac {2 \sqrt {a} f^x x^3 \log ^3(f)}{a+b f^{2 x}}-\frac {3 i x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {i x^3 \log ^3(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {i x^3 \log ^3(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {3 i x \log (f) (-2+x \log (f)) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x \log (f) (-2+x \log (f)) \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i x \log (f) \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i x \log (f) \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}}{4 a^{3/2} \log ^4(f)} \]

[In]

Integrate[(f^x*x^3)/(a + b*f^(2*x))^2,x]

[Out]

((2*Sqrt[a]*f^x*x^3*Log[f]^3)/(a + b*f^(2*x)) - ((3*I)*x^2*Log[f]^2*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b]
+ (I*x^3*Log[f]^3*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((3*I)*x^2*Log[f]^2*Log[1 + (I*Sqrt[b]*f^x)/Sqrt
[a]])/Sqrt[b] - (I*x^3*Log[f]^3*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((3*I)*x*Log[f]*(-2 + x*Log[f])*Po
lyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((3*I)*x*Log[f]*(-2 + x*Log[f])*PolyLog[2, (I*Sqrt[b]*f^x)/Sqr
t[a]])/Sqrt[b] - ((6*I)*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*x*Log[f]*PolyLog[3, ((-I)*Sqr
t[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*x*Log[f]*PolyLog[3,
(I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*PolyLog[4
, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b])/(4*a^(3/2)*Log[f]^4)

Maple [F]

\[\int \frac {f^{x} x^{3}}{\left (a +b \,f^{2 x}\right )^{2}}d x\]

[In]

int(f^x*x^3/(a+b*f^(2*x))^2,x)

[Out]

int(f^x*x^3/(a+b*f^(2*x))^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.10 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {2 \, b f^{x} x^{3} \log \left (f\right )^{3} + 3 \, {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, -f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right )}{4 \, {\left (a b^{2} f^{2 \, x} \log \left (f\right )^{4} + a^{2} b \log \left (f\right )^{4}\right )}} \]

[In]

integrate(f^x*x^3/(a+b*f^(2*x))^2,x, algorithm="fricas")

[Out]

1/4*(2*b*f^x*x^3*log(f)^3 + 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (a*x^2*log(f)^2 - 2*a*x*lo
g(f))*sqrt(-b/a))*dilog(f^x*sqrt(-b/a)) - 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (a*x^2*log(f
)^2 - 2*a*x*log(f))*sqrt(-b/a))*dilog(-f^x*sqrt(-b/a)) - ((b*x^3*log(f)^3 - 3*b*x^2*log(f)^2)*f^(2*x)*sqrt(-b/
a) + (a*x^3*log(f)^3 - 3*a*x^2*log(f)^2)*sqrt(-b/a))*log(f^x*sqrt(-b/a) + 1) + ((b*x^3*log(f)^3 - 3*b*x^2*log(
f)^2)*f^(2*x)*sqrt(-b/a) + (a*x^3*log(f)^3 - 3*a*x^2*log(f)^2)*sqrt(-b/a))*log(-f^x*sqrt(-b/a) + 1) + 6*(b*f^(
2*x)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4, f^x*sqrt(-b/a)) - 6*(b*f^(2*x)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4
, -f^x*sqrt(-b/a)) - 6*((b*x*log(f) - b)*f^(2*x)*sqrt(-b/a) + (a*x*log(f) - a)*sqrt(-b/a))*polylog(3, f^x*sqrt
(-b/a)) + 6*((b*x*log(f) - b)*f^(2*x)*sqrt(-b/a) + (a*x*log(f) - a)*sqrt(-b/a))*polylog(3, -f^x*sqrt(-b/a)))/(
a*b^2*f^(2*x)*log(f)^4 + a^2*b*log(f)^4)

Sympy [F]

\[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {f^{x} x^{3}}{2 a^{2} \log {\left (f \right )} + 2 a b f^{2 x} \log {\left (f \right )}} + \frac {\int \left (- \frac {3 f^{x} x^{2}}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x^{3} \log {\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log {\left (f \right )}} \]

[In]

integrate(f**x*x**3/(a+b*f**(2*x))**2,x)

[Out]

f**x*x**3/(2*a**2*log(f) + 2*a*b*f**(2*x)*log(f)) + (Integral(-3*f**x*x**2/(a + b*f**(2*x)), x) + Integral(f**
x*x**3*log(f)/(a + b*f**(2*x)), x))/(2*a*log(f))

Maxima [F]

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

[In]

integrate(f^x*x^3/(a+b*f^(2*x))^2,x, algorithm="maxima")

[Out]

1/2*f^x*x^3/(a*b*f^(2*x)*log(f) + a^2*log(f)) + integrate(1/2*(x^3*log(f) - 3*x^2)*f^x/(a*b*f^(2*x)*log(f) + a
^2*log(f)), x)

Giac [F]

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

[In]

integrate(f^x*x^3/(a+b*f^(2*x))^2,x, algorithm="giac")

[Out]

integrate(f^x*x^3/(b*f^(2*x) + a)^2, x)

Mupad [F(-1)]

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

[In]

int((f^x*x^3)/(a + b*f^(2*x))^2,x)

[Out]

int((f^x*x^3)/(a + b*f^(2*x))^2, x)