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

Optimal. Leaf size=319 \[ -\frac {2 b e^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{d}+\frac {24 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+\frac {24 i b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {24 i b^3 e^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} n^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{3/2}} \]

[Out]

24*I*b^3*e^(3/2)*n^3*arctan(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

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Rubi [A]  time = 0.49, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 2.66, size = 646, normalized size = 2.03 \[ -\frac {-3 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right ) \left (-6 d \left (d+e x^{2/3}\right ) \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;\frac {x^{2/3} e}{d}+1\right )-2 d \log \left (d+e x^{2/3}\right ) \left (-4 e x^{2/3} \left (\sqrt {-\frac {e x^{2/3}}{d}}-1\right )+4 d \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \log \left (\frac {1}{2} \left (\sqrt {-\frac {e x^{2/3}}{d}}+1\right )\right )+\left (d-d \left (-\frac {e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )\right )\right )+2 b^3 d n^3 \left (-9 \left (d+e x^{2/3}\right ) \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \, _5F_4\left (1,1,1,1,\frac {5}{2};2,2,2,2;\frac {x^{2/3} e}{d}+1\right )+9 \left (d+e x^{2/3}\right ) \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \log \left (d+e x^{2/3}\right ) \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;\frac {x^{2/3} e}{d}+1\right )+\left (-6 e x^{2/3} \left (\sqrt {-\frac {e x^{2/3}}{d}}-1\right )+6 d \left (-\frac {e x^{2/3}}{d}\right )^{3/2} \log \left (\frac {1}{2} \left (\sqrt {-\frac {e x^{2/3}}{d}}+1\right )\right )+\left (d-d \left (-\frac {e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )\right ) \log ^2\left (d+e x^{2/3}\right )\right )+6 b d^2 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+2 d^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^3+12 b \sqrt {d} e^{3/2} n x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+12 b d e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{2 d^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(2*b^3*d*n^3*(-9*(d + e*x^(2/3))*(-((e*x^(2/3))/d))^(3/2)*HypergeometricPFQ[{1, 1, 1, 1, 5/2}, {2, 2, 2,
2}, 1 + (e*x^(2/3))/d] + 9*(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]*Log[d + e*x^(2/3)] + Log[d + e*x^(2/3)]^2*(-6*e*(-1 + Sqrt[-((e*x^(2/3))/d)])*x^(2/3) +
6*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)])) - 3*b^2*n^2*(-6*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] - 2*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]) + 12*b*d*e*n*x^(2/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log
[c*(d + e*x^(2/3))^n])^2 + 12*b*Sqrt[d]*e^(3/2)*n*x*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 + 6*b*d^2*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 + 2*d^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3)/(d^2*x)

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{3}}{x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} n^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{3}}{x} + \int \frac {{\left (2 \, b^{3} e n x + 3 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x + 3 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 3 \, {\left ({\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{\frac {1}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right ) + {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x + {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x^{\frac {1}{3}}}{e x^{3} + d x^{\frac {7}{3}}}\,{d x} \]

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]

-b^3*n^3*log(e*x^(2/3) + d)^3/x + integrate(((2*b^3*e*n*x + 3*(b^3*e*log(c) + a*b^2*e)*x + 3*(b^3*d*log(c) + a
*b^2*d)*x^(1/3))*n^2*log(e*x^(2/3) + d)^2 + 3*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)
^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^(1/3))*n*log(e*x^(2/3) + d) + (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*
b*e*log(c) + a^3*e)*x + (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))/(e*x^3 + d*x
^(7/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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