Optimal. Leaf size=95 \[ 3 b n \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {3}{2} \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-3 b^2 n^2 \text {Li}_3\left (\frac {x^{2/3} e}{d}+1\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2454, 2396, 2433, 2374, 6589} \[ 3 b n \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-3 b^2 n^2 \text {PolyLog}\left (3,\frac {e x^{2/3}}{d}+1\right )+\frac {3}{2} \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2433
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx &=\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac {e x^{2/3}}{d}\right )-(3 b e n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac {e x^{2/3}}{d}\right )-(3 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac {e x^{2/3}}{d}\right )+3 b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-\left (3 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac {e x^{2/3}}{d}\right )+3 b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-3 b^2 n^2 \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )\\ \end {align*}
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Mathematica [B] time = 0.12, size = 199, normalized size = 2.09 \[ 2 b n \left (\log (x) \left (\log \left (d+e x^{2/3}\right )-\log \left (\frac {e x^{2/3}}{d}+1\right )\right )-\frac {3}{2} \text {Li}_2\left (-\frac {e x^{2/3}}{d}\right )\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )+\log (x) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+\frac {3}{2} b^2 n^2 \left (-2 \text {Li}_3\left (\frac {x^{2/3} e}{d}+1\right )+2 \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right ) \log \left (d+e x^{2/3}\right )+\log \left (-\frac {e x^{2/3}}{d}\right ) \log ^2\left (d+e x^{2/3}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 148, normalized size = 1.56 \[ \frac {3}{2} \, {\left (\log \left (e x^{\frac {2}{3}} + d\right )^{2} \log \left (-\frac {e x^{\frac {2}{3}} + d}{d} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {e x^{\frac {2}{3}} + d}{d}\right ) \log \left (e x^{\frac {2}{3}} + d\right ) - 2 \, {\rm Li}_{3}(\frac {e x^{\frac {2}{3}} + d}{d})\right )} b^{2} n^{2} + a^{2} \log \relax (x) + 3 \, {\left (b^{2} n \log \relax (c) + a b n\right )} {\left (\log \left (e x^{\frac {2}{3}} + d\right ) \log \left (-\frac {e x^{\frac {2}{3}} + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x^{\frac {2}{3}} + d}{d}\right )\right )} + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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