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

Optimal. Leaf size=1277 \[ \text{result too large to display} \]

[Out]

(568*a*b^2*e^4*n^2*x^(1/3))/(105*d^4) - (16*b^3*e^4*n^3*x^(1/3))/(7*d^4) + (16*b^3*e^3*n^3*x)/(105*d^3) + (137
6*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(105*d^(9/2)) + (((568*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) - (1136*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e
])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(105*d^(9/2)) + (568*b^3*e^4*n^2*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/(105*d^4
) - (32*b^2*e^3*n^2*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(35*d^3) + (8*b^2*e^2*n^2*x^(5/3)*(a + b*Log[c*(d + e/
x^(2/3))^n]))/(35*d^2) - (568*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n
]))/(105*d^(9/2)) - (2*b*e^4*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d^4 + (2*b*e^3*n*x*(a + b*Log[c*(d
+ e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3
)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/3 + (4*b^2*e^(9/2)*n^2*
(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(9/2) - (2*b^3*e^(9/2)*n^3*Log[Sqrt[e]
- Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) - (4*b^2*e^(9/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]
*x^(1/3)])/(-d)^(9/2) + (2*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) + (4*b^3*e^(9/2)*n^3*
Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*Log
[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*Log[Sqrt
[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(9/2) + (8*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt
[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (((568*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, -1 + (2*
Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2) + (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e
]])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) + (4*b^3*e^(9
/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 + (Sqrt[-
d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (2*b*e^5*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))
*x^(2/3)), x])/(3*d^4)

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Rubi [A]  time = 3.16869, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(568*a*b^2*e^4*n^2*x^(1/3))/(105*d^4) - (16*b^3*e^4*n^3*x^(1/3))/(7*d^4) + (16*b^3*e^3*n^3*x)/(105*d^3) + (137
6*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(105*d^(9/2)) + (((568*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) - (1136*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e
])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(105*d^(9/2)) + (568*b^3*e^4*n^2*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/(105*d^4
) - (32*b^2*e^3*n^2*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(35*d^3) + (8*b^2*e^2*n^2*x^(5/3)*(a + b*Log[c*(d + e/
x^(2/3))^n]))/(35*d^2) - (568*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n
]))/(105*d^(9/2)) - (2*b*e^4*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d^4 + (2*b*e^3*n*x*(a + b*Log[c*(d
+ e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3
)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/3 + (4*b^2*e^(9/2)*n^2*
(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(9/2) - (2*b^3*e^(9/2)*n^3*Log[Sqrt[e]
- Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) - (4*b^2*e^(9/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]
*x^(1/3)])/(-d)^(9/2) + (2*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) + (4*b^3*e^(9/2)*n^3*
Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*Log
[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*Log[Sqrt
[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(9/2) + (8*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt
[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (((568*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, -1 + (2*
Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2) + (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e
]])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) + (4*b^3*e^(9
/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 + (Sqrt[-
d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (2*b*e^5*n*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e/x^2)^n])^2/(e + d*x^
2), x], x, x^(1/3)])/d^4

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \operatorname{Subst}\left (\int \left (-\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^4}+\frac{e^2 x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^3}-\frac{e x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^2}+\frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d}+\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^4 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{(2 b e n) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^2}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^3}-\frac{\left (2 b e^4 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^3}-\frac{e x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2}+\frac{x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d}-\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^3 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2}+\frac{x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d}+\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d}-\frac{e \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^2}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^3}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^3}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac{\left (4 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (16 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}-\frac{\left (16 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}-\frac{\left (16 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{e x}-\frac{d x \log \left (\sqrt{e}-\sqrt{-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{e x}-\frac{d x \log \left (\sqrt{e}+\sqrt{-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{64 b^3 e^4 n^3 \sqrt [3]{x}}{35 d^4}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{e}{d^2}+\frac{x^2}{d}+\frac{e^2}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{35 d^2}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \log \left (\sqrt{e}-\sqrt{-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \log \left (\sqrt{e}+\sqrt{-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1328 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \log \left (\sqrt{e}-\sqrt{-d} x\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \log \left (\sqrt{e}-\sqrt{-d} x\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \log \left (\sqrt{e}+\sqrt{-d} x\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \log \left (\sqrt{e}+\sqrt{-d} x\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (16 i b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac{\left (16 i b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac{\left (16 i b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{\sqrt{-d} x}{\sqrt{e}}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{-d} x}{\sqrt{e}}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{e}-\sqrt{-d} x}{2 \sqrt{e}}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{e}+\sqrt{-d} x}{2 \sqrt{e}}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 \sqrt{e}}\right )}{x} \, dx,x,\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 \sqrt{e}}\right )}{x} \, dx,x,\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \text{Li}_2\left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \text{Li}_2\left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ \end{align*}

Mathematica [A]  time = 4.8286, size = 764, normalized size = 0.6 \[ -\frac{b^2 n^2 \left (-a-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )+b n \log \left (d+\frac{e}{x^{2/3}}\right )\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (9 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-9 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}+\frac{b^3 n^3 \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (27 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+2 d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-54 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )+54 e^5 \left (d x^{2/3}+e\right ) \, _5F_4\left (1,1,1,1,\frac{11}{2};2,2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{6 d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{5 d^2}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^{9/2}}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^3+b n x^3 \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*n^3*(54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 1, 11/2}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[
d + e/x^(2/3)]*(-54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log
[d + e/x^(2/3)]*(27*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + 2*d*x^(2/
3)*(e^5 + d^5*Sqrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))))/(6*d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (b^2*
n^2*(-9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[d + e/x^(2/
3)]*(9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + d*x^(2/3)*(e^5 + d^5*S
qrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/x^(2/3))^n]
))/(d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (2*b*e^4*n*x^(1/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n
])^2)/d^4 + (2*b*e^3*n*x*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(
5/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3)*(a - b*n*Log[d +
e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*
Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/d^(9/2) + b*n*x^3*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^
(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + (x^3*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^3)/3

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Maple [A]  time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{3} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x^2*log(c*((d*x + e*x^(1/3))/x)^n)^3 + 3*a*b^2*x^2*log(c*((d*x + e*x^(1/3))/x)^n)^2 + 3*a^2*b*x^2
*log(c*((d*x + e*x^(1/3))/x)^n) + a^3*x^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(x**2*(a+b*ln(c*(d+e/x**(2/3))**n))**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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