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

Optimal. Leaf size=420 \[ -\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 x^2 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{x f^x}{4 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}-\frac{x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b} \log ^2(f)}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{x^2 f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.582367, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2249, 199, 205, 2245, 14, 2282, 4848, 2391, 12, 5143, 2531, 6589} \[ -\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^3(f)}-\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{3 x^2 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{x f^x}{4 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}-\frac{x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b} \log ^2(f)}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b} \log ^3(f)}+\frac{x^2 f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

Antiderivative was successfully verified.

[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 2249

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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

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

Rule 2245

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 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 2282

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 4848

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 2391

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 12

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

Rule 5143

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 2531

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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{f^x x^2}{\left (a+b f^{2 x}\right )^3} \, 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 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{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,f^x\right )}{4 a^2 \log ^3(f)}+\frac{(3 i) \operatorname{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) \operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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) \operatorname{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]  time = 0.490146, size = 353, normalized size = 0.84 \[ \frac{\frac{3 i \left (2 \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 x \log (f) \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+2 x \log (f) \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )}{\sqrt{a} \sqrt{b}}-\frac{8 i \left (-\text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x \log (f) \left (\log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-\log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )\right )}{\sqrt{a} \sqrt{b}}+\frac{4 a x^2 f^x \log ^2(f)}{\left (a+b f^{2 x}\right )^2}+\frac{2 x f^x \log (f) (3 x \log (f)-2)}{a+b f^{2 x}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}}{16 a^2 \log ^3(f)} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.143, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{x}{x}^{2}}{ \left ( a+b{f}^{2\,x} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 1.68545, size = 1747, normalized size = 4.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 - \frac{8 f^{x} x \log{\left (f \right )}}{a + b f^{2 x}}\, 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{x} x^{2}}{{\left (b f^{2 \, x} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)