3.248 \(\int \frac{(a+b \log (c x^n))^3}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=711 \[ -\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{PolyLog}\left (4,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt{e}} \]

[Out]

(x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)) + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - ((a +
b*Log[c*x^n])^3*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + (S
qrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^3*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/
2)*Sqrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b
*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*
x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[
e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e
]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b^3*n^3*Po
lyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[e]*x)/
Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[4, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (
3*b^3*n^3*PolyLog[4, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 0.845976, antiderivative size = 711, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2330, 2318, 2317, 2374, 6589, 2383} \[ -\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{PolyLog}\left (4,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)) + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - ((a +
b*Log[c*x^n])^3*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + (S
qrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^3*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/
2)*Sqrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b
*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*
x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[
e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e
]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b^3*n^3*Po
lyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[e]*x)/
Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[4, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (
3*b^3*n^3*PolyLog[4, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e])

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

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]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{-d e-e^2 x^2} \, dx}{2 d}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^3}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^3}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}-\frac{\left (3 b \sqrt{e} n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{e}-e x} \, dx}{4 (-d)^{3/2}}-\frac{\left (3 b \sqrt{e} n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{e}+e x} \, dx}{4 (-d)^{3/2}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{4 (-d)^{3/2} \sqrt{e}}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{3 b^3 n^3 \text{Li}_4\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{3 b^3 n^3 \text{Li}_4\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [C]  time = 2.19558, size = 1073, normalized size = 1.51 \[ \frac{\frac{i b^3 \left (\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log ^3(x)-\log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) \log ^3(x)+\frac{\sqrt{d} \log ^3(x)}{i \sqrt{e} x+\sqrt{d}}+\frac{\sqrt{e} x \log ^3(x)}{\sqrt{e} x+i \sqrt{d}}-\log ^3(x)-3 \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log ^2(x)+3 \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) \log ^2(x)-3 (\log (x)-2) \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)+3 (\log (x)-2) \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)+6 \text{PolyLog}\left (3,-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)-6 \text{PolyLog}\left (3,\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)-6 \text{PolyLog}\left (3,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+6 \text{PolyLog}\left (3,\frac{i \sqrt{e} x}{\sqrt{d}}\right )-6 \text{PolyLog}\left (4,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+6 \text{PolyLog}\left (4,\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right ) n^3}{\sqrt{e}}+3 b^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac{\log (x) \left (\sqrt{e} x \log (x)+2 i \left (i \sqrt{e} x+\sqrt{d}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )\right )+2 i \left (i \sqrt{e} x+\sqrt{d}\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{i e x+\sqrt{d} \sqrt{e}}+\frac{\log (x) \left (\sqrt{e} x \log (x)-2 i \left (\sqrt{d}-i \sqrt{e} x\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )-2 \left (\sqrt{e} x+i \sqrt{d}\right ) \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}-i e x}-\frac{i \left (\log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right ) \log ^2(x)+2 \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)-2 \text{PolyLog}\left (3,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e}}+\frac{i \left (\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log ^2(x)+2 \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right ) \log (x)-2 \text{PolyLog}\left (3,\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e}}\right ) n^2+3 b \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\frac{\sqrt{e} x \log (x)+i \left (i \sqrt{e} x+\sqrt{d}\right ) \log \left (i \sqrt{d}-\sqrt{e} x\right )}{i e x+\sqrt{d} \sqrt{e}}+\frac{\sqrt{e} x \log (x)+\left (-\sqrt{e} x-i \sqrt{d}\right ) \log \left (\sqrt{e} x+i \sqrt{d}\right )}{\sqrt{d} \sqrt{e}-i e x}-\frac{i \left (\log (x) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )+\text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e}}+\frac{i \left (\log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e}}\right ) n+\frac{2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3}{\sqrt{e}}+\frac{2 \sqrt{d} x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3}{e x^2+d}}{4 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[d]*x*(a - b*n*Log[x] + b*Log[c*x^n])^3)/(d + e*x^2) + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a - b*n*Log[x]
+ b*Log[c*x^n])^3)/Sqrt[e] + 3*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*((Sqrt[e]*x*Log[x] + I*(Sqrt[d] + I*Sqrt[
e]*x)*Log[I*Sqrt[d] - Sqrt[e]*x])/(Sqrt[d]*Sqrt[e] + I*e*x) + (Sqrt[e]*x*Log[x] + ((-I)*Sqrt[d] - Sqrt[e]*x)*L
og[I*Sqrt[d] + Sqrt[e]*x])/(Sqrt[d]*Sqrt[e] - I*e*x) - (I*(Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + PolyLog[2,
((-I)*Sqrt[e]*x)/Sqrt[d]]))/Sqrt[e] + (I*(Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + PolyLog[2, (I*Sqrt[e]*x)/Sqr
t[d]]))/Sqrt[e]) + 3*b^2*n^2*(a - b*n*Log[x] + b*Log[c*x^n])*((Log[x]*(Sqrt[e]*x*Log[x] + (2*I)*(Sqrt[d] + I*S
qrt[e]*x)*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]]) + (2*I)*(Sqrt[d] + I*Sqrt[e]*x)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]]
)/(Sqrt[d]*Sqrt[e] + I*e*x) + (Log[x]*(Sqrt[e]*x*Log[x] - (2*I)*(Sqrt[d] - I*Sqrt[e]*x)*Log[1 - (I*Sqrt[e]*x)/
Sqrt[d]]) - 2*(I*Sqrt[d] + Sqrt[e]*x)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e] - I*e*x) - (I*(Log[x
]^2*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + 2*Log[x]*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - 2*PolyLog[3, ((-I)*Sqrt[e
]*x)/Sqrt[d]]))/Sqrt[e] + (I*(Log[x]^2*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 2*Log[x]*PolyLog[2, (I*Sqrt[e]*x)/Sqrt
[d]] - 2*PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]]))/Sqrt[e]) + (I*b^3*n^3*(-Log[x]^3 + (Sqrt[d]*Log[x]^3)/(Sqrt[d] +
I*Sqrt[e]*x) + (Sqrt[e]*x*Log[x]^3)/(I*Sqrt[d] + Sqrt[e]*x) - 3*Log[x]^2*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + Log[
x]^3*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 3*Log[x]^2*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] - Log[x]^3*Log[1 + (I*Sqrt[e]*
x)/Sqrt[d]] - 3*(-2 + Log[x])*Log[x]*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 3*(-2 + Log[x])*Log[x]*PolyLog[2,
(I*Sqrt[e]*x)/Sqrt[d]] - 6*PolyLog[3, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 6*Log[x]*PolyLog[3, ((-I)*Sqrt[e]*x)/Sqrt[d]
] + 6*PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]] - 6*Log[x]*PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]] - 6*PolyLog[4, ((-I)*Sqrt
[e]*x)/Sqrt[d]] + 6*PolyLog[4, (I*Sqrt[e]*x)/Sqrt[d]]))/Sqrt[e])/(4*d^(3/2))

________________________________________________________________________________________

Maple [F]  time = 7.904, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{ \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{3}}{\left (d + e x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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