\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^3}{x^4} \, dx\) [488]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1376 b^3 e^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{105 d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {1408 i b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac {2 b e^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{3 d^4} \]

[Out]

-16/105*b^3*e^3*n^3/d^3/x+16/7*b^3*e^4*n^3/d^4/x^(1/3)+1376/105*b^3*e^(9/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2)
)/d^(9/2)-1408/105*I*b^3*e^(9/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/d^(9/2)-8/35*b^2*e^2*n^2*(a+b*ln(c*(d+e
*x^(2/3))^n))/d^2/x^(5/3)+32/35*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(2/3))^n))/d^3/x-568/105*b^2*e^4*n^2*(a+b*ln(c*(d
+e*x^(2/3))^n))/d^4/x^(1/3)-1408/105*b^2*e^(9/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n
))/d^(9/2)-2/7*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d/x^(7/3)+2/5*b*e^2*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d^2/x^(
5/3)-2/3*b*e^3*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d^3/x+2*b*e^4*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d^4/x^(1/3)-1/3*(
a+b*ln(c*(d+e*x^(2/3))^n))^3/x^3-2816/105*b^3*e^(9/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2
)+I*x^(1/3)*e^(1/2)))/d^(9/2)-1408/105*I*b^3*e^(9/2)*n^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/d^
(9/2)+2/3*b*e^5*n*Unintegrable((a+b*ln(c*(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/d^4

Rubi [N/A]

Not integrable

Time = 1.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]

[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*} \text {integral}& = 3 \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1385\) vs. \(2(632)=1264\).

Time = 7.58 (sec) , antiderivative size = 1385, normalized size of antiderivative = 57.71 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\frac {1}{210} \left (-\frac {60 b e n \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 x^{7/3}}+\frac {84 b e^2 n \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 x^{5/3}}-\frac {140 b e^3 n \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 x}+\frac {420 b e^4 n \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 \sqrt [3]{x}}+\frac {420 b e^{9/2} n \arctan \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}{d^{9/2}}-\frac {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}{x^3}-\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}{x^3}-\frac {2 b^3 n^3 \left (1376 e^3 \left (d+e x^{2/3}\right )^{3/2} \left (\frac {e x^{2/3}}{d+e x^{2/3}}\right )^{3/2} x^2 \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right )+\sqrt {d} \left (16 e^3 \left (d-15 e x^{2/3}\right ) x^2+8 \left (3 d^2 e^2 x^{4/3}-12 d e^3 x^2+71 e^4 x^{8/3}\right ) \log \left (d+e x^{2/3}\right )+\left (30 d^3 e x^{2/3}-42 d^2 e^2 x^{4/3}+70 d e^3 x^2-210 e^4 x^{8/3}\right ) \log ^2\left (d+e x^{2/3}\right )+35 d^4 \log ^3\left (d+e x^{2/3}\right )\right )+210 e^4 \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} x^{8/3} \left (8 \sqrt {d} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^{2/3}}\right )+\log \left (d+e x^{2/3}\right ) \left (4 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^{2/3}}\right )+\sqrt {d+e x^{2/3}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right ) \log \left (d+e x^{2/3}\right )\right )\right )+\frac {352 d^{3/2} e^4 x^{8/3} \left (4 \sqrt {e x^{2/3}} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right )-\sqrt {-d} \sqrt {-\frac {e x^{2/3}}{d}} \left (2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right )-4 \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )+\log ^2\left (1+\frac {e x^{2/3}}{d}\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e x^{2/3}}{d}}\right )\right )\right )}{(-d)^{3/2}}\right )}{d^{9/2} x^3}+b^2 e^5 n^2 \sqrt [3]{x} \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 ) \left (\frac {2816 \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right )}{d^{9/2} \sqrt {d+e x^{2/3}} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}}}-\frac {120 \log \left (d+e x^{2/3}\right )}{d e^4 x^{8/3}}-\frac {210 \log ^2\left (d+e x^{2/3}\right )}{e^5 x^{10/3}}+\frac {24 \left (-2+7 \log \left (d+e x^{2/3}\right )\right )}{d^2 e^3 x^2}-\frac {8 \left (-24+35 \log \left (d+e x^{2/3}\right )\right )}{d^3 e^2 x^{4/3}}+\frac {8 \left (-142+105 \log \left (d+e x^{2/3}\right )\right )}{d^4 e x^{2/3}}-\frac {840 \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right )}{(-d)^{9/2} \sqrt {e x^{2/3}}}-\frac {210 \left (2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right )-4 \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )+\log ^2\left (1+\frac {e x^{2/3}}{d}\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e x^{2/3}}{d}}\right )\right )}{d^5 \sqrt {-\frac {e x^{2/3}}{d}}}\right )\right ) \]

[In]

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

[Out]

((-60*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^(7/3)) + (84*b*e^2*n*(a - b*n*Lo
g[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d^2*x^(5/3)) - (140*b*e^3*n*(a - b*n*Log[d + e*x^(2/3)] + b
*Log[c*(d + e*x^(2/3))^n])^2)/(d^3*x) + (420*b*e^4*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])
^2)/(d^4*x^(1/3)) + (420*b*e^(9/2)*n*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)/d^(9/2) - (210*b*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)/x^3 - (70*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3)/x^3 - (2*b^3*n^3*(1376*e^3*(
d + e*x^(2/3))^(3/2)*((e*x^(2/3))/(d + e*x^(2/3)))^(3/2)*x^2*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + Sqrt[d]*(16
*e^3*(d - 15*e*x^(2/3))*x^2 + 8*(3*d^2*e^2*x^(4/3) - 12*d*e^3*x^2 + 71*e^4*x^(8/3))*Log[d + e*x^(2/3)] + (30*d
^3*e*x^(2/3) - 42*d^2*e^2*x^(4/3) + 70*d*e^3*x^2 - 210*e^4*x^(8/3))*Log[d + e*x^(2/3)]^2 + 35*d^4*Log[d + e*x^
(2/3)]^3) + 210*e^4*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(8/3)*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2
}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2
, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + (3
52*d^(3/2)*e^4*x^(8/3)*(4*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x
^(2/3))/d]) - Sqrt[-d]*Sqrt[-((e*x^(2/3))/d)]*(2*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*
x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Log[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]
/2])))/(-d)^(3/2)))/(d^(9/2)*x^3) + b^2*e^5*n^2*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^
n])*((2816*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(d^(9/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]
) - (120*Log[d + e*x^(2/3)])/(d*e^4*x^(8/3)) - (210*Log[d + e*x^(2/3)]^2)/(e^5*x^(10/3)) + (24*(-2 + 7*Log[d +
e*x^(2/3)]))/(d^2*e^3*x^2) - (8*(-24 + 35*Log[d + e*x^(2/3)]))/(d^3*e^2*x^(4/3)) + (8*(-142 + 105*Log[d + e*x
^(2/3)]))/(d^4*e*x^(2/3)) - (840*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d
]))/((-d)^(9/2)*Sqrt[e*x^(2/3)]) - (210*(2*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3
))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Log[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/
(d^5*Sqrt[-((e*x^(2/3))/d)])))/210

Maple [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}}{x^{4}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{4}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^4} \,d x \]

[In]

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

[Out]

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