\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^3}{x^2} \, dx\) [487]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\frac {24 i b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {24 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+\frac {24 i b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac {2 b e^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{d} \]

[Out]

24*I*b^3*e^(3/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/d^(3/2)+24*b^2*e^(3/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/
2))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^(3/2)-6*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d/x^(1/3)-(a+b*ln(c*(d+e*x^(2/3)
)^n))^3/x+48*b^3*e^(3/2)*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^(3/2)
+24*I*b^3*e^(3/2)*n^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/d^(3/2)-2*b*e^2*n*Unintegrable((a+b*l
n(c*(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/d

Rubi [N/A]

Not integrable

Time = 0.32 (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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

[In]

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

[Out]

((24*I)*b^3*e^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(3/2) + (48*b^3*e^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(
1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(3/2) + (24*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[e]*x
^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^(3/2) - (6*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x
^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/x + ((24*I)*b^3*e^(3/2)*n^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] +
I*Sqrt[e]*x^(1/3))])/d^(3/2) - (6*b*e^2*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

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^4} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+(6 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^2 \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}{x}+(6 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^2}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d \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}{x}+\frac {(6 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^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = -\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 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 )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (48 b^3 e^3 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} \\ & = \frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (48 b^3 e^{5/2} 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^{3/2}} \\ & = \frac {24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (48 b^3 e^2 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^2} \\ & = \frac {24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} 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^{3/2}}+\frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (48 b^3 e^2 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^2} \\ & = \frac {24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} 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^{3/2}}+\frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (48 i b^3 e^{3/2} 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^{3/2}} \\ & = \frac {24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} 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^{3/2}}+\frac {24 b^2 e^{3/2} 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^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+\frac {24 i b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{d^{3/2}}-\frac {\left (6 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}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1158\) vs. \(2(319)=638\).

Time = 6.63 (sec) , antiderivative size = 1158, normalized size of antiderivative = 48.25 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {6 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 \sqrt [3]{x}}-\frac {6 b e^{3/2} 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^{3/2}}-\frac {3 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}-\frac {\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}+\frac {3 b^2 e n^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 ) \left (-\frac {16 \sqrt {d+e x^{2/3}} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right )}{d^{3/2}}-\frac {8 \log \left (d+e x^{2/3}\right )}{d}-\frac {2 \log ^2\left (d+e x^{2/3}\right )}{e x^{2/3}}-\frac {8 \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 )}{(-d)^{3/2}}-\frac {2 \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 )}{d}\right )}{2 \sqrt [3]{x}}+\frac {b^3 n^3 \left (48 \sqrt {-d^2} e \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} x^{2/3} \, _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 )-12 d \sqrt {-d^2} \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right )-24 \sqrt {d} \left (e x^{2/3}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \log \left (d+e x^{2/3}\right )+24 \sqrt {-d^2} e \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} x^{2/3} \, _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 ) \log \left (d+e x^{2/3}\right )-6 \sqrt {-d^2} e x^{2/3} \log ^2\left (d+e x^{2/3}\right )+6 \sqrt {-d} \left (d+e x^{2/3}\right )^{3/2} \left (\frac {e x^{2/3}}{d+e x^{2/3}}\right )^{3/2} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right ) \log ^2\left (d+e x^{2/3}\right )+\frac {d^{5/2} \log ^3\left (d+e x^{2/3}\right )}{\sqrt {-d}}+24 \sqrt {d} \left (e x^{2/3}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )+24 d \sqrt {-d^2} \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \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 )-6 d \sqrt {-d^2} \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \log ^2\left (1+\frac {e x^{2/3}}{d}\right )+24 d \sqrt {-d^2} \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e x^{2/3}}{d}}\right )\right )}{\sqrt {-d} d^{3/2} x} \]

[In]

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

[Out]

(-6*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^(1/3)) - (6*b*e^(3/2)*n*ArcTan[(Sq
rt[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^(3/2) - (3*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 - (a - b*n*Log[d + e*x^(2/3)] + b*Lo
g[c*(d + e*x^(2/3))^n])^3/x + (3*b^2*e*n^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-16*Sqr
t[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/d^(3/2) - (8*Log[d + e
*x^(2/3)])/d - (2*Log[d + e*x^(2/3)]^2)/(e*x^(2/3)) - (8*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Lo
g[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/(-d)^(3/2) - (2*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*Po
lyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/d))/(2*x^(1/3)) + (b^3*n^3*(48*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e
*x^(2/3))]*x^(2/3)*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] - 12*d*Sqrt[-d^
2]*(-((e*x^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 24*Sqrt[d]*(e*x^(2/3))^(3/2)*ArcTanh[Sqrt[
e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^(2/3)] + 24*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*Hypergeome
tricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - 6*Sqrt[-d^2]*e*x^(2/3)*Log[d + e*
x^(2/3)]^2 + 6*Sqrt[-d]*(d + e*x^(2/3))^(3/2)*((e*x^(2/3))/(d + e*x^(2/3)))^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^
(2/3)]]*Log[d + e*x^(2/3)]^2 + (d^(5/2)*Log[d + e*x^(2/3)]^3)/Sqrt[-d] + 24*Sqrt[d]*(e*x^(2/3))^(3/2)*ArcTanh[
Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[1 + (e*x^(2/3))/d] + 24*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e
*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] - 6*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[1 + (e*x^(2/3))/d]^2 +
24*d*Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/(Sqrt[-d]*d^(3/2)*x)

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^3/x^2,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 = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^2} \,d x \]

[In]

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

[Out]

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