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

Optimal. Leaf size=793 \[ \text{result too large to display} \]

[Out]

(4504*a*b^2*d^4*n^2*x^(1/3))/(315*e^4) - (3475504*b^3*d^4*n^3*x^(1/3))/(99225*e^4) + (637984*b^3*d^3*n^3*x)/(2
97675*e^3) - (221344*b^3*d^2*n^3*x^(5/3))/(496125*e^2) + (3088*b^3*d*n^3*x^(7/3))/(27783*e) - (16*b^3*n^3*x^3)
/729 + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(99225*e^(9/2)) - (((4504*I)/315)*b^3*d^(9/
2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(9/2) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*
Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(315*e^(9/2)) + (4504*b^3*d^4*n^2*x^(1/3)*Log[c*(d + e*x^(2/3)
)^n])/(315*e^4) - (1984*b^2*d^3*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(945*e^3) + (1144*b^2*d^2*n^2*x^(5/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(1575*e^2) - (128*b^2*d*n^2*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(441*e
) + (8*b^2*n^2*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/81 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(315*e^(9/2)) - (2*b*d^4*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e
^4 + (2*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3)
)^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) - (2*b*n*x^3*(a + b*Log[c*(d + e
*x^(2/3))^n])^2)/9 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/3 - (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, 1
- (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2) + (2*b*d^5*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^
n])^2/((d + e*x^(2/3))*x^(2/3)), x])/(3*e^4)

________________________________________________________________________________________

Rubi [A]  time = 3.00639, 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 = {} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(4504*a*b^2*d^4*n^2*x^(1/3))/(315*e^4) - (3475504*b^3*d^4*n^3*x^(1/3))/(99225*e^4) + (637984*b^3*d^3*n^3*x)/(2
97675*e^3) - (221344*b^3*d^2*n^3*x^(5/3))/(496125*e^2) + (3088*b^3*d*n^3*x^(7/3))/(27783*e) - (16*b^3*n^3*x^3)
/729 + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(99225*e^(9/2)) - (((4504*I)/315)*b^3*d^(9/
2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(9/2) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*
Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(315*e^(9/2)) + (4504*b^3*d^4*n^2*x^(1/3)*Log[c*(d + e*x^(2/3)
)^n])/(315*e^4) - (1984*b^2*d^3*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(945*e^3) + (1144*b^2*d^2*n^2*x^(5/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(1575*e^2) - (128*b^2*d*n^2*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(441*e
) + (8*b^2*n^2*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/81 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(315*e^(9/2)) - (2*b*d^4*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e
^4 + (2*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3)
)^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) - (2*b*n*x^3*(a + b*Log[c*(d + e
*x^(2/3))^n])^2)/9 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/3 - (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, 1
- (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2) + (2*b*d^5*n*Defer[Subst][Defer[Int][(a + b*Log[c*(d +
e*x^2)^n])^2/(d + e*x^2), x], x, x^(1/3)])/e^4

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+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+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+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \operatorname{Subst}\left (\int \frac{x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \operatorname{Subst}\left (\int \left (\frac{d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5}-\frac{d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^4}+\frac{d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^3}-\frac{d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^2}+\frac{x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e}-\frac{d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b n) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )-\frac{\left (2 b d^4 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (2 b d^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (2 b d^2 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac{(2 b d n) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{1}{7} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac{1}{9} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{1}{7} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac{d^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac{d x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac{d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{e}-\frac{d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac{d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac{d x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac{d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac{1}{9} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac{d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac{d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac{d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac{d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{1}{9} \left (8 b^2 n^2\right ) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^2}-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e}-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e}\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (8 b^3 d^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac{\left (8 b^3 d^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac{\left (8 b^3 d^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac{\left (8 b^3 d^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (8 b^3 d^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{1}{63} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{49} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac{1}{81} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}+\frac{4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{1}{63} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac{1}{49} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac{\left (16 b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^{7/2}}+\frac{\left (16 b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^{7/2}}+\frac{\left (16 b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^{7/2}}+\frac{\left (16 b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^{7/2}}+\frac{\left (16 b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^{7/2}}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac{1}{81} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \left (\frac{d^4}{e^5}-\frac{d^3 x^2}{e^4}+\frac{d^2 x^4}{e^3}-\frac{d x^6}{e^2}+\frac{x^8}{e}-\frac{d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac{3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac{637984 b^3 d^3 n^3 x}{297675 e^3}-\frac{221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac{3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac{16}{729} b^3 n^3 x^3-\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{315 e^{9/2}}+\frac{4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{81 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{63 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{49 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}+2 \frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac{3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac{637984 b^3 d^3 n^3 x}{297675 e^3}-\frac{221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac{3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac{16}{729} b^3 n^3 x^3+\frac{3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{99225 e^{9/2}}-\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac{4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac{3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac{637984 b^3 d^3 n^3 x}{297675 e^3}-\frac{221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac{3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac{16}{729} b^3 n^3 x^3+\frac{3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{99225 e^{9/2}}-\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac{4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (16 i b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{9 e^{9/2}}-\frac{\left (16 i b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{7 e^{9/2}}-\frac{\left (16 i b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{5 e^{9/2}}-\frac{\left (16 i b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{3 e^{9/2}}-\frac{\left (16 i b^3 d^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{e^{9/2}}\\ &=\frac{4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac{3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac{637984 b^3 d^3 n^3 x}{297675 e^3}-\frac{221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac{3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac{16}{729} b^3 n^3 x^3+\frac{3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{99225 e^{9/2}}-\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac{4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac{1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac{128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac{8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{315 e^{9/2}}-\frac{2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac{2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac{2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac{2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac{2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac{4504 i b^3 d^{9/2} n^3 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{315 e^{9/2}}+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ \end{align*}

Mathematica [A]  time = 9.0109, size = 3146, normalized size = 3.97 \[ \text{Result too large to show} \]

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*x^(1/3)*(32*d^4 - 32*d^4*Sqrt[1 - (d + e*x^(2/3))/d] + 128*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(
2/3)) - 192*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 + 128*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/
3))^3 - 32*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 + 1584*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1
, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 4536*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}
, (d + e*x^(2/3))/d] + 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/
d] - 864*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 3024*d^3
*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 3780*d^3*(d + e*x^(2
/3))*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x^(2/3))*Hyperge
ometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 240*d^4*Log[d + e*x^(2/3)] + 240*d^4*Sqrt[1
- (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 672*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3
)] + 576*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] - 96*d*Sqrt[1 - (d + e*x^(2/3))/
d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)] - 48*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]
- 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 864
*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 302
4*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 37
80*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 1
890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] +
284*d^4*Log[d + e*x^(2/3)]^2 - 284*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 668*d^3*Sqrt[1 - (d
+ e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^2 - 552*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*L
og[d + e*x^(2/3)]^2 + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 - 68*Sqrt[1 - (
d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1
, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 945*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}
, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 70*d^4*Log[d + e*x^(2/3)]^3 + 70*d^4*Sqrt[1 - (d + e*x^(2/
3))/d]*Log[d + e*x^(2/3)]^3 - 280*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^3 + 420*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^3 - 280*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d +
e*x^(2/3))^3*Log[d + e*x^(2/3)]^3 + 70*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^3 + 1
512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(1 + 3*Log[d + e*x^(2/3)] +
Log[d + e*x^(2/3)]^2) - 144*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(6
+ 11*Log[d + e*x^(2/3)] + 3*Log[d + e*x^(2/3)]^2)))/(210*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) + (b^2*n^2*x^(1/3)*
(-120*d^4 + 120*d^4*Sqrt[1 - (d + e*x^(2/3))/d] - 336*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3)) + 288*d^
2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 - 48*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3 - 24*Sqrt
[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (
d + e*x^(2/3))/d] + 432*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] -
1512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x
^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 945*d^3*(d + e*x^(2/3))*Hypergeomet
ricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 284*d^4*Log[d + e*x^(2/3)] - 284*d^4*Sqrt[1 - (d + e*x
^(2/3))/d]*Log[d + e*x^(2/3)] + 668*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)] - 552*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e
*x^(2/3))^3*Log[d + e*x^(2/3)] - 68*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)] - 1890*d^
3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 945*d^3*(d +
e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 105*d^4*Log[d + e*
x^(2/3)]^2 + 105*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 420*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d
+ e*x^(2/3))*Log[d + e*x^(2/3)]^2 + 630*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^
2 - 420*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 + 105*Sqrt[1 - (d + e*x^(2/3))/d]
*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 + 756*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d +
e*x^(2/3))/d]*(3 + 2*Log[d + e*x^(2/3)]) - 72*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d
+ e*x^(2/3))/d]*(11 + 6*Log[d + e*x^(2/3)]))*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/(10
5*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) - (2*b*d^4*n*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3)
)^n]))^2)/e^4 + (2*b*d^3*n*x*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(3*e^3) - (2*b*d^
2*n*x^(5/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*
(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/(7*e) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2)/e^(9/2) + b*n*x^3*Log[d + e*x^(2/3)]*(a +
b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n]))^2 + (x^3*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d +
e*x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/9

________________________________________________________________________________________

Maple [A]  time = 0.341, 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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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