3.531 \(\int \frac{(a+b \log (c (d+\frac{e}{x^{2/3}})^n))^3}{x^4} \, dx\)

Optimal. Leaf size=783 \[ \text{result too large to display} \]

[Out]

(16*b^3*n^3)/(729*x^3) - (3088*b^3*d*n^3)/(27783*e*x^(7/3)) + (221344*b^3*d^2*n^3)/(496125*e^2*x^(5/3)) - (637
984*b^3*d^3*n^3)/(297675*e^3*x) + (3475504*b^3*d^4*n^3)/(99225*e^4*x^(1/3)) + (3475504*b^3*d^(9/2)*n^3*ArcTan[
(Sqrt[d]*x^(1/3))/Sqrt[e]])/(99225*e^(9/2)) + (((4504*I)/315)*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]
]^2)/e^(9/2) - (9008*b^3*d^(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))])/(315*e^(9/2)) - (8*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(81*x^3) + (128*b^2*d*n^2*(a + b*Log
[c*(d + e/x^(2/3))^n]))/(441*e*x^(7/3)) - (1144*b^2*d^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(1575*e^2*x^(5/3
)) + (1984*b^2*d^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(945*e^3*x) - (4504*b^2*d^4*n^2*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^4*x^(1/3)) - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^(9/2)) + (2*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(9*x^3) - (2*b*d*n*(a + b*Log[c*(d + e
/x^(2/3))^n])^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(
a + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e^4*x^(1/3)) -
(a + b*Log[c*(d + e/x^(2/3))^n])^3/(3*x^3) + (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt
[e] - I*Sqrt[d]*x^(1/3))])/e^(9/2) + (2*b*d^5*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3
))*x^(2/3)), x])/(3*e^4)

________________________________________________________________________________________

Rubi [A]  time = 3.62489, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4,x]

[Out]

(16*b^3*n^3)/(729*x^3) - (3088*b^3*d*n^3)/(27783*e*x^(7/3)) + (221344*b^3*d^2*n^3)/(496125*e^2*x^(5/3)) - (637
984*b^3*d^3*n^3)/(297675*e^3*x) + (3475504*b^3*d^4*n^3)/(99225*e^4*x^(1/3)) + (3475504*b^3*d^(9/2)*n^3*ArcTan[
(Sqrt[d]*x^(1/3))/Sqrt[e]])/(99225*e^(9/2)) + (((4504*I)/315)*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]
]^2)/e^(9/2) - (9008*b^3*d^(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))])/(315*e^(9/2)) - (8*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(81*x^3) + (128*b^2*d*n^2*(a + b*Log
[c*(d + e/x^(2/3))^n]))/(441*e*x^(7/3)) - (1144*b^2*d^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(1575*e^2*x^(5/3
)) + (1984*b^2*d^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(945*e^3*x) - (4504*b^2*d^4*n^2*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^4*x^(1/3)) - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^(9/2)) + (2*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(9*x^3) - (2*b*d*n*(a + b*Log[c*(d + e
/x^(2/3))^n])^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(
a + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e^4*x^(1/3)) -
(a + b*Log[c*(d + e/x^(2/3))^n])^3/(3*x^3) + (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt
[e] - I*Sqrt[d]*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/(e + d*
x^2), x], x, x^(1/3)])/e^4

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^3}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{\left (d+\frac{e}{x^2}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b e n) \operatorname{Subst}\left (\int \left (\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e x^{10}}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^2 x^8}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^3 x^6}-\frac{d^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^4 x^4}+\frac{d^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^5 x^2}-\frac{d^5 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^5 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^{10}} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (2 b d^4 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^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+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (2 b d^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (2 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^6} \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac{(2 b d n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^8} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}-\frac{1}{7} \left (8 b^2 d 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^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^4 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^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (8 b^2 d^3 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^6} \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac{\left (8 b^2 d^2 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^8} \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac{1}{9} \left (8 b^2 e 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^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}-\frac{1}{7} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^8}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^6}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^3 x^4}-\frac{d^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^4 x^2}+\frac{d^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^4 \left (e+d 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+\frac{e}{x^2}\right )^n\right )}{e x^2}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e \left (e+d 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{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^4}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 \left (e+d 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{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^6}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^4}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^3 x^2}-\frac{d^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^3 \left (e+d 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{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^{10}}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^8}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^3 x^6}-\frac{d^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^4 x^4}+\frac{d^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^5 x^2}-\frac{d^5 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^5 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}+\frac{1}{9} \left (8 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, 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+\frac{e}{x^2}\right )^n\right )}{e+d 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+\frac{e}{x^2}\right )^n\right )}{e+d 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+\frac{e}{x^2}\right )^n\right )}{e+d 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+\frac{e}{x^2}\right )^n\right )}{e+d 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+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{7 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{5 e^2}-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{9 e}-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{7 e}\\ &=-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}+\frac{1}{63} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{49} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^{10}} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac{\left (16 b^3 d^5 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 )}{9 e^3}-\frac{\left (16 b^3 d^5 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 e^3}-\frac{\left (16 b^3 d^5 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 e^3}-\frac{\left (16 b^3 d^5 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 e^3}-\frac{\left (16 b^3 d^5 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 )}{e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac{1}{81} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}+\frac{1}{63} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{49} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (16 b^3 d^{9/2} 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 )}{9 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 )}{e^{7/2}}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d 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{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac{1}{81} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^{10} \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{256 b^3 d n^3}{3087 e x^{7/3}}+\frac{2288 b^3 d^2 n^3}{7875 e^2 x^{5/3}}-\frac{3968 b^3 d^3 n^3}{2835 e^3 x}+\frac{9008 b^3 d^4 n^3}{315 e^4 \sqrt [3]{x}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}+\frac{1}{81} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d 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}{e+d 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}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac{\left (16 b^3 d^{9/2} 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 )}{9 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac{\left (16 b^3 d^{9/2} 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 )}{e^{7/2}}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{27 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e}\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{3088 b^3 d n^3}{27783 e x^{7/3}}+\frac{7472 b^3 d^2 n^3}{18375 e^2 x^{5/3}}-\frac{26704 b^3 d^3 n^3}{14175 e^3 x}+\frac{30992 b^3 d^4 n^3}{945 e^4 \sqrt [3]{x}}+\frac{9008 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{315 e^{9/2}}+\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{315 e^{9/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}-\frac{\left (16 i b^3 d^{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 )}{9 e^{9/2}}-\frac{\left (16 i b^3 d^{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 e^{9/2}}-\frac{\left (16 i b^3 d^{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 e^{9/2}}-\frac{\left (16 i b^3 d^{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 e^{9/2}}-\frac{\left (16 i b^3 d^{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 )}{e^{9/2}}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e^2}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e^2}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e}\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{3088 b^3 d n^3}{27783 e x^{7/3}}+\frac{221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac{206128 b^3 d^3 n^3}{99225 e^3 x}+\frac{161824 b^3 d^4 n^3}{4725 e^4 \sqrt [3]{x}}+\frac{30992 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{945 e^{9/2}}+\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{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 )}{315 e^{9/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b d^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 )}{e^4}+\frac{\left (16 b^3 d^5 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 )}{9 e^5}+\frac{\left (16 b^3 d^5 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 e^5}+\frac{\left (16 b^3 d^5 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 e^5}+\frac{\left (16 b^3 d^5 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 e^5}+\frac{\left (16 b^3 d^5 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 )}{e^5}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e^3}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e^3}+\frac{\left (16 b^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e^2}\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{3088 b^3 d n^3}{27783 e x^{7/3}}+\frac{221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac{637984 b^3 d^3 n^3}{297675 e^3 x}+\frac{1151968 b^3 d^4 n^3}{33075 e^4 \sqrt [3]{x}}+\frac{161824 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{4725 e^{9/2}}+\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{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 )}{315 e^{9/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{4504 i b^3 d^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\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+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{49 e^4}-\frac{\left (16 b^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e^3}\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{3088 b^3 d n^3}{27783 e x^{7/3}}+\frac{221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac{637984 b^3 d^3 n^3}{297675 e^3 x}+\frac{3475504 b^3 d^4 n^3}{99225 e^4 \sqrt [3]{x}}+\frac{1151968 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{33075 e^{9/2}}+\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{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 )}{315 e^{9/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{4504 i b^3 d^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\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+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac{\left (16 b^3 d^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{81 e^4}\\ &=\frac{16 b^3 n^3}{729 x^3}-\frac{3088 b^3 d n^3}{27783 e x^{7/3}}+\frac{221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac{637984 b^3 d^3 n^3}{297675 e^3 x}+\frac{3475504 b^3 d^4 n^3}{99225 e^4 \sqrt [3]{x}}+\frac{3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{99225 e^{9/2}}+\frac{4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{315 e^{9/2}}-\frac{9008 b^3 d^{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 )}{315 e^{9/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac{128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac{1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac{1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac{4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac{4504 b^2 d^{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 )}{315 e^{9/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac{2 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac{2 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac{2 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac{2 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac{4504 i b^3 d^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\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+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ \end{align*}

Mathematica [A]  time = 9.0079, size = 2858, normalized size = 3.65 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4,x]

[Out]

-(b^3*n^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] - 86
4*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))*Hy
pergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e/x^(2/3))/d] + 1890*d^3*(d + e/x^(2/3))*Hypergeometric
PFQ[{-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)] + 57
6*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)] - 3024*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)] + 3780*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)] - 1890*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*Log[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 + 1512*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*L
og[d + e/x^(2/3)] + 3*Log[d + e/x^(2/3)]^2)))/(210*e^4*Sqrt[1 - (d + e/x^(2/3))/d]*x^(1/3)) - (2*b*d*n*(a + b*
(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*(-(n*Log[d + e/x^(2
/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d
+ e/x^(2/3))^n]))^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2/3))^n]))^2)/(e
^4*x^(1/3)) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/
x^(2/3))^n]))^2)/e^(9/2) - (b*n*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*x^3) + 9*b^2*n^2*(a + b*(-(n*Log[d + e/x^(2/3)]) + Log[c*(d + e/x^(2
/3))^n]))*(-Log[(e + d*x^(2/3))/x^(2/3)]^2/(9*x^3) + (-9800*e^(9/2) + 28800*d*e^(7/2)*x^(2/3) - 72072*d^2*e^(5
/2)*x^(4/3) + 208320*d^3*e^(3/2)*x^2 - 1418760*d^4*Sqrt[e]*x^(8/3) + 1418760*d^(9/2)*x^3*ArcTan[Sqrt[e]/(Sqrt[
d]*x^(1/3))] + 44100*e^(9/2)*Log[d + e/x^(2/3)] - 56700*d*e^(7/2)*x^(2/3)*Log[d + e/x^(2/3)] + 79380*d^2*e^(5/
2)*x^(4/3)*Log[d + e/x^(2/3)] - 132300*d^3*e^(3/2)*x^2*Log[d + e/x^(2/3)] + 396900*d^4*Sqrt[e]*x^(8/3)*Log[d +
e/x^(2/3)] + 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] - 99225*(-d)^(9/2)*x^3*
Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2 - 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] +
99225*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2 + 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)
]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (
Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))
/Sqrt[e])] + 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]] + 396900*(-
d)^(9/2)*x^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] - 198450*(-d)^(9/2)*x^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(
1/3))/(2*Sqrt[e])] + 198450*(-d)^(9/2)*x^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 396900*(-d)^(9/2)*
x^3*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(893025*e^(9/2)*x^3))

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Maple [A]  time = 0.351, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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((a+b*ln(c*(d+e/x^(2/3))^n))^3/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,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 (\frac{b^{3} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,x, algorithm="fricas")

[Out]

integral((b^3*log(c*((d*x + e*x^(1/3))/x)^n)^3 + 3*a*b^2*log(c*((d*x + e*x^(1/3))/x)^n)^2 + 3*a^2*b*log(c*((d*
x + e*x^(1/3))/x)^n) + a^3)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/x**(2/3))**n))**3/x**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,x, algorithm="giac")

[Out]

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