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

Optimal. Leaf size=318 \[ -\frac{2 b e^2 n \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{d}+\frac{24 i b^3 e^{3/2} n^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac{24 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac{6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+\frac{24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{48 b^3 e^{3/2} n^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}} \]

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

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Rubi [A]  time = 0.486985, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx &=3 \operatorname{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) \operatorname{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) \operatorname{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) \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}-\frac{\left (6 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}{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 ) \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}+\frac{\left (24 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 )}{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 ) \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}-\frac{\left (48 b^3 e^3 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}\\ &=\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 ) \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}-\frac{\left (48 b^3 e^{5/2} 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^{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 ) \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}+\frac{\left (48 b^3 e^2 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^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 ) \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}-\frac{\left (48 b^3 e^2 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^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 ) \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}+\frac{\left (48 i b^3 e^{3/2} 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^{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 ) \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}\\ \end{align*}

Mathematica [A]  time = 7.16765, size = 1028, normalized size = 3.23 \[ \frac{b^3 \left (\frac{d^{5/2} \log ^3\left (d+e x^{2/3}\right )}{\sqrt{-d}}+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} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{d+e x^{2/3}}}\right ) \log ^2\left (d+e x^{2/3}\right )-6 \sqrt{-d^2} e x^{2/3} \log ^2\left (d+e x^{2/3}\right )-24 \sqrt{d} \left (e x^{2/3}\right )^{3/2} \tanh ^{-1}\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 )-12 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right )-6 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log ^2\left (\frac{x^{2/3} e}{d}+1\right )+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 )+24 \sqrt{d} \left (e x^{2/3}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e x^{2/3}}}{\sqrt{-d}}\right ) \log \left (\frac{x^{2/3} e}{d}+1\right )+24 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \log \left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right ) \log \left (\frac{x^{2/3} e}{d}+1\right )+24 d \sqrt{-d^2} \left (-\frac{e x^{2/3}}{d}\right )^{3/2} \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{e x^{2/3}}{d}}\right )\right ) n^3}{\sqrt{-d} d^{3/2} x}-\frac{3 b^2 \left (-3 d \left (d+e x^{2/3}\right ) \, _4F_3\left (1,1,1,\frac{5}{2};2,2,2;\frac{x^{2/3} e}{d}+1\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2}-d \log \left (d+e x^{2/3}\right ) \left (4 d \log \left (\frac{1}{2} \left (\sqrt{-\frac{e x^{2/3}}{d}}+1\right )\right ) \left (-\frac{e x^{2/3}}{d}\right )^{3/2}+\left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-4 e \left (\sqrt{-\frac{e x^{2/3}}{d}}-1\right ) x^{2/3}\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 ) n^2}{d^2 x}-\frac{6 b e^{3/2} \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 n}{d^{3/2}}-\frac{3 b \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 n}{x}-\frac{6 b e \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 n}{d \sqrt [3]{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} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*b^2*n^2*(-3*d*(d + e*x^(2/3))*(-((e*x^(2/3))/d))^(3/2)*HypergeometricPFQ[{1, 1, 1, 5/2}, {2, 2, 2}, 1 + (e
*x^(2/3))/d] - d*Log[d + e*x^(2/3)]*(-4*e*(-1 + Sqrt[-((e*x^(2/3))/d)])*x^(2/3) + 4*d*(-((e*x^(2/3))/d))^(3/2)
*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2] + (d - d*(-((e*x^(2/3))/d))^(3/2))*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^2*x) - (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[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Lo
g[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*Log[c*(d + e*x^(2/3))^n])^3/x + (b^3*n^3*(48*Sqrt[-d^2]*e*Sq
rt[(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)*HypergeometricPFQ[{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]))/(Sq
rt[-d]*d^(3/2)*x)

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Maple [A]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

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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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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