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

Optimal. Leaf size=860 \[ \text{result too large to display} \]

[Out]

(x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^2)/((1 + (-1)^(1
/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e
^(1/3)*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n
])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1
/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*(a + b*Log[c*x^n])^2*Log[1 - ((
-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log
[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3
)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*b^2*n^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3
)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)
*b^2*n^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((4*I)*Sqrt[3]*b*n
*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)
^(1/3)*b^2*n^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*
PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -(
(e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + ((4*I)*Sqrt[3]*b^2*n^2*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]
)/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^
(1/3))^4*d^(5/3)*e^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.771024, antiderivative size = 860, 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, 2391, 2374, 6589} \[ -\frac{2 b^2 \text{PolyLog}\left (2,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b^2 \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 b^2 \text{PolyLog}\left (3,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac{4 i \sqrt{3} b^2 \text{PolyLog}\left (3,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{4 b^2 \text{PolyLog}\left (3,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac{2 b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac{4 b \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 i \sqrt{3} b \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{4 b \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac{\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^2)/((1 + (-1)^(1
/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e
^(1/3)*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n
])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1
/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*(a + b*Log[c*x^n])^2*Log[1 - ((
-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log
[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3
)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*b^2*n^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3
)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)
*b^2*n^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((4*I)*Sqrt[3]*b*n
*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)
^(1/3)*b^2*n^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*
PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -(
(e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + ((4*I)*Sqrt[3]*b^2*n^2*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]
)/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^
(1/3))^4*d^(5/3)*e^(1/3))

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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2}-\frac{2 (-1)^{5/6} \sqrt{3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}+\frac{2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=\frac{2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac{2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{\left (2 (-1)^{5/6} \sqrt{3}\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{(4 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (4 \sqrt [3]{-1} b n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (4 i \sqrt{3} b n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 (-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 i \sqrt{3} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{4 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}-\frac{\left (4 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}-\frac{\left (2 \sqrt [3]{-1} b^2 n^2\right ) \int \frac{\log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (4 \sqrt [3]{-1} b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (2 (-1)^{2/3} b^2 n^2\right ) \int \frac{\log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (4 i \sqrt{3} b^2 n^2\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 (-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 b^2 n^2 \text{Li}_2\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 (-1)^{2/3} b^2 n^2 \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 i \sqrt{3} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \sqrt [3]{-1} b^2 n^2 \text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{4 b^2 n^2 \text{Li}_3\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{4 i \sqrt{3} b^2 n^2 \text{Li}_3\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{4 \sqrt [3]{-1} b^2 n^2 \text{Li}_3\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}\\ \end{align*}

Mathematica [A]  time = 6.15188, size = 1379, normalized size = 1.6 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2)/(3*d*(d + e*x^3)) + (2*ArcTan[(-d^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/
3))]*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2)/(3*Sqrt[3]*d^(5/3)*e^(1/3)) + (2*(a + b*(-(n*Log[x]) + Log[c*x^n]))
^2*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) - ((a + b*(-(n*Log[x]) + Log[c*x^n]))^2*Log[d^(2/3) - d^(1/3)
*e^(1/3)*x + e^(2/3)*x^2])/(9*d^(5/3)*e^(1/3)) + 2*b*n*(a + b*(-(n*Log[x]) + Log[c*x^n]))*(-((-1 + (-1)^(1/3))
*((-((-1)^(1/3)/d^(1/3)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1))*Log[x] + ((-1)^(1/3)*Log[-((-1)^(2/3)*d^(1/3
)) - e^(1/3)*x])/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) + ((-1)^(1/3)*((d^(-1/3) - (d^(1/3) + e^(1/3
)*x)^(-1))*Log[x] - Log[d^(1/3) + e^(1/3)*x]/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) - (Log[x]/(e^(1/
3)*((-1)^(1/3)*d^(1/3) - e^(1/3)*x)) - (-(((-1)^(2/3)*Log[x])/d^(1/3)) + ((-1)^(2/3)*Log[d^(1/3) + (-1)^(2/3)*
e^(1/3)*x])/d^(1/3))/e^(1/3))/(3*(1 + (-1)^(1/3))^2*d^(4/3)) + (2*(-1)^(1/3)*(Log[x]*Log[1 + (e^(1/3)*x)/d^(1/
3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]*Log[1 - ((-1)^(
1/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3))
- (2*(-1 + (-1)^(1/3))*(Log[x]*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/
d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3))) + b^2*n^2*(((-1)^(1/3)*(Log[x]*((e^(1/3)*x*Log[x])/(d^(1/3
) + e^(1/3)*x) - 2*Log[1 + (e^(1/3)*x)/d^(1/3)]) - 2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^
2*d^(5/3)*e^(1/3)) - ((-1 + (-1)^(1/3))*(Log[x]*((-((-1)^(1/3)/d^(1/3)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1
))*Log[x] + (2*(-1)^(1/3)*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/d^(1/3)) + (2*(-1)^(1/3)*PolyLog[2, ((-1)^(
1/3)*e^(1/3)*x)/d^(1/3)])/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d^(4/3)*e^(1/3)) - (Log[x]*((-1)^(2/3)*e^(1/3)*x*Log
[x] - 2*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)]) - 2*(d^(1/3) + (-1)^(2/3)*e^
(1/3)*x)*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*(1 + (-1)^(1/3))^2*d^(4/3)*(-((-1)^(1/3)*d^(2/3)*e^
(1/3)) + d^(1/3)*e^(2/3)*x)) + (2*(-1)^(1/3)*(Log[x]^2*Log[1 + (e^(1/3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, -((e
^(1/3)*x)/d^(1/3))] - 2*PolyLog[3, -((e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[
x]^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - 2*PolyLog
[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(-1 + (-1)^(1/3))*(Log[x]^2*
Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + 2*Log[x]*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))] - 2*PolyLog[3
, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)))

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Maple [F]  time = 4.986, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( e{x}^{3}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^6 + 2*d*e*x^3 + d^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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