\(\int \frac {f^x x^2}{(a+b f^{2 x})^3} \, dx\) [53]

Optimal result

Integrand size = 18, antiderivative size = 420 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)} \]

[Out]

-1/4*f^x*x/a^2/(a+b*f^(2*x))/ln(f)^2+1/4*f^x*x^2/a/(a+b*f^(2*x))^2/ln(f)+3/8*f^x*x^2/a^2/(a+b*f^(2*x))/ln(f)+1
/4*arctan(f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^3/b^(1/2)-x*arctan(f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^2/b^(1/2)+3
/8*x^2*arctan(f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)/b^(1/2)+1/2*I*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)
^3/b^(1/2)-3/8*I*x*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^2/b^(1/2)-1/2*I*polylog(2,I*f^x*b^(1/2)/a^(
1/2))/a^(5/2)/ln(f)^3/b^(1/2)+3/8*I*x*polylog(2,I*f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^2/b^(1/2)+3/8*I*polylog(3
,-I*f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^3/b^(1/2)-3/8*I*polylog(3,I*f^x*b^(1/2)/a^(1/2))/a^(5/2)/ln(f)^3/b^(1/2
)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2281, 205, 211, 2277, 14, 2320, 4940, 2438, 12, 5251, 2611, 6724} \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {3 x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 x^2 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}-\frac {x f^x}{4 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac {x^2 f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

[In]

Int[(f^x*x^2)/(a + b*f^(2*x))^3,x]

[Out]

ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(4*a^(5/2)*Sqrt[b]*Log[f]^3) - (f^x*x)/(4*a^2*(a + b*f^(2*x))*Log[f]^2) - (x*Arc
Tan[(Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^2) + (f^x*x^2)/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x*x^2
)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]) + ((I/2)*P
olyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^3) - (((3*I)/8)*x*PolyLog[2, ((-I)*Sqrt[b]*f^x)
/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^2) - ((I/2)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^3)
+ (((3*I)/8)*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^2) + (((3*I)/8)*PolyLog[3, ((-I)*
Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^3) - (((3*I)/8)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*S
qrt[b]*Log[f]^3)

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 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 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-2 \int x \left (\frac {f^x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-2 \int \left (\frac {f^x x}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 \int \frac {f^x x}{a+b f^{2 x}} \, dx}{4 a^2 \log (f)}-\frac {\int \frac {f^x x}{\left (a+b f^{2 x}\right )^2} \, dx}{2 a \log (f)}-\frac {3 \int x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log (f)} \\ & = -\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {3 \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{4 a^2 \log (f)}+\frac {\int \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}{2 a \log (f)}-\frac {(3 i) \int x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {(3 i) \int x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log (f)} \\ & = -\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\int \frac {f^x}{a+b f^{2 x}} \, dx}{4 a^2 \log ^2(f)}+\frac {(3 i) \int \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 \int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{5/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,f^x\right )}{4 a^2 \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(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 )}{8 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {\text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {(3 i) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {f^x x}{4 a^2 \left (a+b f^{2 x}\right ) \log ^2(f)}-\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{4 a \left (a+b f^{2 x}\right )^2 \log (f)}+\frac {3 f^x x^2}{8 a^2 \left (a+b f^{2 x}\right ) \log (f)}+\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log (f)}+\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}-\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{5/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^2(f)}+\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} \log ^3(f)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.84 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {\frac {4 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {4 a f^x x^2 \log ^2(f)}{\left (a+b f^{2 x}\right )^2}+\frac {2 f^x x \log (f) (-2+3 x \log (f))}{a+b f^{2 x}}-\frac {8 i \left (x \log (f) \left (\log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-\log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {b}}+\frac {3 i \left (x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-2 x \log (f) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+2 x \log (f) \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-2 \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {b}}}{16 a^2 \log ^3(f)} \]

[In]

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

[Out]

((4*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (4*a*f^x*x^2*Log[f]^2)/(a + b*f^(2*x))^2 + (2*f^x*x*Log
[f]*(-2 + 3*x*Log[f]))/(a + b*f^(2*x)) - ((8*I)*(x*Log[f]*(Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]] - Log[1 + (I*Sqrt[
b]*f^x)/Sqrt[a]]) - PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] + PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]]))/(Sqrt[a]*Sq
rt[b]) + ((3*I)*(x^2*Log[f]^2*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]] - x^2*Log[f]^2*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]]
- 2*x*Log[f]*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] + 2*x*Log[f]*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]] + 2*Poly
Log[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] - 2*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]]))/(Sqrt[a]*Sqrt[b]))/(16*a^2*Log[f]
^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (298) = 596\).

Time = 0.27 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.87 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {2 \, {\left (3 \, b^{2} x^{2} \log \left (f\right )^{2} - 2 \, b^{2} x \log \left (f\right )\right )} f^{3 \, x} + 2 \, {\left (5 \, a b x^{2} \log \left (f\right )^{2} - 2 \, a b x \log \left (f\right )\right )} f^{x} + 2 \, {\left ({\left (3 \, b^{2} x \log \left (f\right ) - 4 \, b^{2}\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x \log \left (f\right ) - 4 \, a b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x \log \left (f\right ) - 4 \, a^{2}\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 2 \, {\left ({\left (3 \, b^{2} x \log \left (f\right ) - 4 \, b^{2}\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x \log \left (f\right ) - 4 \, a b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x \log \left (f\right ) - 4 \, a^{2}\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) + 2 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt {-\frac {b}{a}}\right ) - 2 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt {-\frac {b}{a}}\right ) - {\left ({\left (3 \, b^{2} x^{2} \log \left (f\right )^{2} - 8 \, b^{2} x \log \left (f\right )\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x^{2} \log \left (f\right )^{2} - 8 \, a b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x^{2} \log \left (f\right )^{2} - 8 \, a^{2} x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (3 \, b^{2} x^{2} \log \left (f\right )^{2} - 8 \, b^{2} x \log \left (f\right )\right )} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, {\left (3 \, a b x^{2} \log \left (f\right )^{2} - 8 \, a b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (3 \, a^{2} x^{2} \log \left (f\right )^{2} - 8 \, a^{2} x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) - 6 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, {\left (b^{2} f^{4 \, x} \sqrt {-\frac {b}{a}} + 2 \, a b f^{2 \, x} \sqrt {-\frac {b}{a}} + a^{2} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right )}{16 \, {\left (a^{2} b^{3} f^{4 \, x} \log \left (f\right )^{3} + 2 \, a^{3} b^{2} f^{2 \, x} \log \left (f\right )^{3} + a^{4} b \log \left (f\right )^{3}\right )}} \]

[In]

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

[Out]

1/16*(2*(3*b^2*x^2*log(f)^2 - 2*b^2*x*log(f))*f^(3*x) + 2*(5*a*b*x^2*log(f)^2 - 2*a*b*x*log(f))*f^x + 2*((3*b^
2*x*log(f) - 4*b^2)*f^(4*x)*sqrt(-b/a) + 2*(3*a*b*x*log(f) - 4*a*b)*f^(2*x)*sqrt(-b/a) + (3*a^2*x*log(f) - 4*a
^2)*sqrt(-b/a))*dilog(f^x*sqrt(-b/a)) - 2*((3*b^2*x*log(f) - 4*b^2)*f^(4*x)*sqrt(-b/a) + 2*(3*a*b*x*log(f) - 4
*a*b)*f^(2*x)*sqrt(-b/a) + (3*a^2*x*log(f) - 4*a^2)*sqrt(-b/a))*dilog(-f^x*sqrt(-b/a)) + 2*(b^2*f^(4*x)*sqrt(-
b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*log(2*b*f^x + 2*a*sqrt(-b/a)) - 2*(b^2*f^(4*x)*sqrt(-b/a) +
2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*log(2*b*f^x - 2*a*sqrt(-b/a)) - ((3*b^2*x^2*log(f)^2 - 8*b^2*x*log(
f))*f^(4*x)*sqrt(-b/a) + 2*(3*a*b*x^2*log(f)^2 - 8*a*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (3*a^2*x^2*log(f)^2 - 8*
a^2*x*log(f))*sqrt(-b/a))*log(f^x*sqrt(-b/a) + 1) + ((3*b^2*x^2*log(f)^2 - 8*b^2*x*log(f))*f^(4*x)*sqrt(-b/a)
+ 2*(3*a*b*x^2*log(f)^2 - 8*a*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (3*a^2*x^2*log(f)^2 - 8*a^2*x*log(f))*sqrt(-b/a
))*log(-f^x*sqrt(-b/a) + 1) - 6*(b^2*f^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*polylog(3
, f^x*sqrt(-b/a)) + 6*(b^2*f^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*polylog(3, -f^x*sqr
t(-b/a)))/(a^2*b^3*f^(4*x)*log(f)^3 + 2*a^3*b^2*f^(2*x)*log(f)^3 + a^4*b*log(f)^3)

Sympy [F]

\[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^3} \, dx=\frac {f^{3 x} \left (3 b x^{2} \log {\left (f \right )} - 2 b x\right ) + f^{x} \left (5 a x^{2} \log {\left (f \right )} - 2 a x\right )}{8 a^{4} \log {\left (f \right )}^{2} + 16 a^{3} b f^{2 x} \log {\left (f \right )}^{2} + 8 a^{2} b^{2} f^{4 x} \log {\left (f \right )}^{2}} + \frac {\int \frac {2 f^{x}}{a + b f^{2 x}}\, dx + \int \left (- \frac {8 f^{x} x \log {\left (f \right )}}{a + b f^{2 x}}\right )\, dx + \int \frac {3 f^{x} x^{2} \log {\left (f \right )}^{2}}{a + b f^{2 x}}\, dx}{8 a^{2} \log {\left (f \right )}^{2}} \]

[In]

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

[Out]

(f**(3*x)*(3*b*x**2*log(f) - 2*b*x) + f**x*(5*a*x**2*log(f) - 2*a*x))/(8*a**4*log(f)**2 + 16*a**3*b*f**(2*x)*l
og(f)**2 + 8*a**2*b**2*f**(4*x)*log(f)**2) + (Integral(2*f**x/(a + b*f**(2*x)), x) + Integral(-8*f**x*x*log(f)
/(a + b*f**(2*x)), x) + Integral(3*f**x*x**2*log(f)**2/(a + b*f**(2*x)), x))/(8*a**2*log(f)**2)

Maxima [F]

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

[In]

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

[Out]

1/8*((3*b*x^2*log(f) - 2*b*x)*f^(3*x) + (5*a*x^2*log(f) - 2*a*x)*f^x)/(a^2*b^2*f^(4*x)*log(f)^2 + 2*a^3*b*f^(2
*x)*log(f)^2 + a^4*log(f)^2) + integrate(1/8*(3*x^2*log(f)^2 - 8*x*log(f) + 2)*f^x/(a^2*b*f^(2*x)*log(f)^2 + a
^3*log(f)^2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((f^x*x^2)/(a + b*f^(2*x))^3,x)

[Out]

int((f^x*x^2)/(a + b*f^(2*x))^3, x)