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

Optimal. Leaf size=632 \[ -\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{105 d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {1408 i b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac {2 b e^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{3 d^4} \]

[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 [A]
time = 1.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \end {gather*}

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 \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*}

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Mathematica [A]
time = 2.53, size = 803, normalized size = 1.27 \begin {gather*} \frac {35 b^3 n^3 \left (54 e^4 \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;1+\frac {e x^{2/3}}{d}\right )+\log \left (d+e x^{2/3}\right ) \left (54 d e^3 \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;1+\frac {e x^{2/3}}{d}\right )+\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;1+\frac {e x^{2/3}}{d}\right )-2 d \left (d^4+d e^3 \left (-\frac {e x^{2/3}}{d}\right )^{3/2} x^2\right ) \log \left (d+e x^{2/3}\right )\right )\right )\right )+\frac {210 b^2 n^2 \left (-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;1+\frac {e x^{2/3}}{d}\right )+\log \left (d+e x^{2/3}\right ) \left (9 e^5 \left (d+e x^{2/3}\right ) x^{10/3} \, _3F_2\left (1,1,\frac {11}{2};2,2;1+\frac {e x^{2/3}}{d}\right )+d \left (d^5 \sqrt {-\frac {e x^{2/3}}{d}}+e^5 x^{10/3}\right ) \log \left (d+e x^{2/3}\right )\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 )}{d \sqrt {-\frac {e x^{2/3}}{d}}}-60 b d^4 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+84 b d^3 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 )^2-140 b d^2 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+420 b d 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+420 b \sqrt {d} 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 )^2-210 b d^5 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 )^2-70 d^5 \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}{210 d^5 x^3} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}}{x^{4}}\, dx\]

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

-1/3*b^3*n^3*log(x^(2/3)*e + d)^3/x^3 + integrate(1/3*((2*b^3*n*x*e + 9*(b^3*log(c) + a*b^2)*x*e + 9*(b^3*d*lo
g(c) + a*b^2*d)*x^(1/3))*n^2*log(x^(2/3)*e + d)^2 + 3*(b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)
*x*e + 9*((b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*x*e + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^(1/3))
*n*log(x^(2/3)*e + d) + 3*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^(1/3))/(x^5*e + d
*x^(13/3)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^(2/3)*e + d)^n*c) + a)^3/x^4, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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