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

Optimal. Leaf size=631 \[ \frac{2 b e^5 n \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{3 d^4}-\frac{1408 i b^3 e^{9/2} n^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{105 d^{9/2}}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}-\frac{1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^{9/2}}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}-\frac{16 b^3 e^3 n^3}{105 d^3 x}-\frac{1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{105 d^{9/2}}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{2816 b^3 e^{9/2} n^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}} \]

[Out]

(-16*b^3*e^3*n^3)/(105*d^3*x) + (16*b^3*e^4*n^3)/(7*d^4*x^(1/3)) + (1376*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/
3))/Sqrt[d]])/(105*d^(9/2)) - (((1408*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(9/2) - (
2816*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(105*d^
(9/2)) - (8*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^2*x^(5/3)) + (32*b^2*e^3*n^2*(a + b*Log[c*(d +
e*x^(2/3))^n]))/(35*d^3*x) - (568*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^4*x^(1/3)) - (1408*b^2
*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^(9/2)) - (2*b*e*n*(a +
b*Log[c*(d + e*x^(2/3))^n])^2)/(7*d*x^(7/3)) + (2*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(5*d^2*x^(5/3))
- (2*b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*d^3*x) + (2*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(
d^4*x^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/(3*x^3) - (((1408*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (2*
Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(9/2) + (2*b*e^5*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^n])^2/
((d + e*x^(2/3))*x^(2/3)), x])/(3*d^4)

________________________________________________________________________________________

Rubi [A]  time = 1.87028, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]

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*e^3*n^3)/(105*d^3*x) + (16*b^3*e^4*n^3)/(7*d^4*x^(1/3)) + (1376*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/
3))/Sqrt[d]])/(105*d^(9/2)) - (((1408*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(9/2) - (
2816*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(105*d^
(9/2)) - (8*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^2*x^(5/3)) + (32*b^2*e^3*n^2*(a + b*Log[c*(d +
e*x^(2/3))^n]))/(35*d^3*x) - (568*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^4*x^(1/3)) - (1408*b^2
*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^(9/2)) - (2*b*e*n*(a +
b*Log[c*(d + e*x^(2/3))^n])^2)/(7*d*x^(7/3)) + (2*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(5*d^2*x^(5/3))
- (2*b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*d^3*x) + (2*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(
d^4*x^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/(3*x^3) - (((1408*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (2*
Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(9/2) + (2*b*e^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)])/d^4

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+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+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+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+e x^2\right )^n\right )\right )^2}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+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+e x^2\right )^n\right )\right )^2}{d x^8}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^2 x^6}+\frac{e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^3 x^4}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^4 x^2}+\frac{e^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^8} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^6} \, dx,x,\sqrt [3]{x}\right )}{d^2}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )}{d^3}-\frac{\left (2 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{8 b^2 e^{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 )}{d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^6}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^4}+\frac{e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 x^2}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^4}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^2}+\frac{e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^2}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}+\frac{\left (16 b^3 e^6 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 )}{d^4}\\ &=-\frac{8 b^2 e^{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 )}{d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{7 d^2}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{7 d^3}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{5 d^3}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac{\left (8 b^2 e^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 d^4}-\frac{\left (8 b^2 e^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 d^4}-\frac{\left (8 b^2 e^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 d^4}+\frac{\left (16 b^3 e^{11/2} 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 )}{d^{9/2}}\\ &=-\frac{8 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}-\frac{\left (16 b^3 e^5 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 )}{d^5}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (16 b^3 e^6 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 d^4}+\frac{\left (16 b^3 e^6 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 d^4}+\frac{\left (16 b^3 e^6 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 d^4}\\ &=-\frac{16 b^3 e^3 n^3}{105 d^3 x}+\frac{64 b^3 e^4 n^3}{35 d^4 \sqrt [3]{x}}+\frac{1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{8 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{9/2}}-\frac{16 b^3 e^{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 )}{d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^3}+\frac{\left (16 b^3 e^5 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 )}{d^5}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}+\frac{\left (16 b^3 e^{11/2} 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 d^{9/2}}+\frac{\left (16 b^3 e^{11/2} 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 d^{9/2}}+\frac{\left (16 b^3 e^{11/2} 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 d^{9/2}}\\ &=-\frac{16 b^3 e^3 n^3}{105 d^3 x}+\frac{16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac{1328 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{105 d^{9/2}}-\frac{16 b^3 e^{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 )}{d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac{\left (2 b e^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 )}{d^4}-\frac{\left (16 i b^3 e^{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 )}{d^{9/2}}-\frac{\left (16 b^3 e^5 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 d^5}-\frac{\left (16 b^3 e^5 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 d^5}-\frac{\left (16 b^3 e^5 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 d^5}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}\\ &=-\frac{16 b^3 e^3 n^3}{105 d^3 x}+\frac{16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{105 d^{9/2}}-\frac{2816 b^3 e^{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 )}{105 d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac{8 i b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{d^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^5 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 d^5}+\frac{\left (16 b^3 e^5 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 d^5}+\frac{\left (16 b^3 e^5 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 d^5}\\ &=-\frac{16 b^3 e^3 n^3}{105 d^3 x}+\frac{16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{105 d^{9/2}}-\frac{2816 b^3 e^{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 )}{105 d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac{8 i b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{d^{9/2}}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (16 i b^3 e^{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 d^{9/2}}-\frac{\left (16 i b^3 e^{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 d^{9/2}}-\frac{\left (16 i b^3 e^{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 d^{9/2}}\\ &=-\frac{16 b^3 e^3 n^3}{105 d^3 x}+\frac{16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{105 d^{9/2}}-\frac{1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{105 d^{9/2}}-\frac{2816 b^3 e^{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 )}{105 d^{9/2}}-\frac{8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac{32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac{568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac{1408 b^2 e^{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 )}{105 d^{9/2}}-\frac{2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac{2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac{1408 i b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{105 d^{9/2}}+\frac{\left (2 b e^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 )}{d^4}\\ \end{align*}

Mathematica [A]  time = 2.8917, size = 803, normalized size = 1.27 \[ \frac{-70 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^5-210 b n \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^5-60 b e n x^{2/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^4+84 b e^2 n x^{4/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^3-140 b e^3 n x^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^2+420 b e^4 n x^{8/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d+420 b e^{9/2} n x^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \sqrt{d}+35 b^3 n^3 \left (54 \left (d+e x^{2/3}\right ) \sqrt{-\frac{e x^{2/3}}{d}} x^{8/3} \, _5F_4\left (1,1,1,1,\frac{11}{2};2,2,2,2;\frac{x^{2/3} e}{d}+1\right ) e^4+\log \left (d+e x^{2/3}\right ) \left (54 d \left (d+e x^{2/3}\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2} x^2 \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right ) e^3+\log \left (d+e x^{2/3}\right ) \left (27 e^4 \left (d+e x^{2/3}\right ) \sqrt{-\frac{e x^{2/3}}{d}} x^{8/3} \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{x^{2/3} e}{d}+1\right )-2 d \left (d^4+e^3 \left (-\frac{e x^{2/3}}{d}\right )^{3/2} x^2 d\right ) \log \left (d+e x^{2/3}\right )\right )\right )\right )+\frac{210 b^2 n^2 \left (\log \left (d+e x^{2/3}\right ) \left (9 \left (d+e x^{2/3}\right ) x^{10/3} \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{x^{2/3} e}{d}+1\right ) e^5+d \left (\sqrt{-\frac{e x^{2/3}}{d}} d^5+e^5 x^{10/3}\right ) \log \left (d+e x^{2/3}\right )\right )-9 e^5 \left (d+e x^{2/3}\right ) x^{10/3} \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right )\right ) \left (-a+b n \log \left (d+e x^{2/3}\right )-b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{\sqrt{-\frac{e x^{2/3}}{d}} d}}{210 d^5 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*b^3*n^3*(54*e^4*(d + e*x^(2/3))*Sqrt[-((e*x^(2/3))/d)]*x^(8/3)*HypergeometricPFQ[{1, 1, 1, 1, 11/2}, {2, 2
, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(54*d*e^3*(d + e*x^(2/3))*(-((e*x^(2/3))/d))^(3/2)*x^2*Hyperg
eometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(27*e^4*(d + e*x^(2/3))*Sqrt[-
((e*x^(2/3))/d)]*x^(8/3)*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + (e*x^(2/3))/d] - 2*d*(d^4 + d*e^3*(-((e*x
^(2/3))/d))^(3/2)*x^2)*Log[d + e*x^(2/3)]))) + (210*b^2*n^2*(-9*e^5*(d + e*x^(2/3))*x^(10/3)*HypergeometricPFQ
[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(9*e^5*(d + e*x^(2/3))*x^(10/3)*Hypergeom
etricPFQ[{1, 1, 11/2}, {2, 2}, 1 + (e*x^(2/3))/d] + d*(d^5*Sqrt[-((e*x^(2/3))/d)] + e^5*x^(10/3))*Log[d + e*x^
(2/3)]))*(-a + b*n*Log[d + e*x^(2/3)] - b*Log[c*(d + e*x^(2/3))^n]))/(d*Sqrt[-((e*x^(2/3))/d)]) - 60*b*d^4*e*n
*x^(2/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + 84*b*d^3*e^2*n*x^(4/3)*(a - b*n*Log[d +
e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 140*b*d^2*e^3*n*x^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e
*x^(2/3))^n])^2 + 420*b*d*e^4*n*x^(8/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + 420*b*Sq
rt[d]*e^(9/2)*n*x^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]
)^2 - 210*b*d^5*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 70*d^5*(a -
b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3)/(210*d^5*x^3)

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Maple [A]  time = 0.364, 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c)^3 + 3*a*b^2*log((e*x^(2/3) + d)^n*c)^2 + 3*a^2*b*log((e*x^(2/3) + d)^n*
c) + 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c) + a)^3/x^4, x)