Optimal. Leaf size=238 \[ \frac {b e^3 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e x^{2/3}}\right )}{d^3}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}} \]
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Rubi [A] time = 0.50, antiderivative size = 261, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac {b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right )}{d^3}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac {b e^3 n \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2454
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^3} \, dx &=\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,x^{2/3}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+(b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx,x,x^{2/3}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+(b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x^{2/3}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+\frac {(b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x^{2/3}\right )}{d}-\frac {(b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{d}\\ &=-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac {(b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{d^2}+\frac {\left (b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x^{2/3}\right )}{d^2}+\frac {\left (b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{2 d}\\ &=-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x^{2/3}\right )}{d^3}-\frac {\left (b e^3 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x^{2/3}\right )}{d^3}+\frac {\left (b^2 e n^2\right ) \operatorname {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+e x^{2/3}\right )}{2 d}-\frac {\left (b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x^{2/3}\right )}{d^3}\\ &=-\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {\left (b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )}{d^3}\\ &=-\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}+\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {b^2 e^3 n^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 264, normalized size = 1.11 \[ -\frac {\frac {e x^{2/3} \left (3 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-6 b e^2 n x^{4/3} \left (\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+b n \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right )\right )+3 e^2 x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b d e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-2 b^2 e^2 n^2 x^{4/3} \left (3 \log \left (d+e x^{2/3}\right )-2 \log (x)\right )+b^2 e n^2 x^{2/3} \left (-3 e x^{2/3} \log \left (d+e x^{2/3}\right )+3 d+2 e x^{2/3} \log (x)\right )\right )}{d^3}+3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{2}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{2} n^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2}}{2 \, x^{2}} + \int \frac {2 \, {\left (b^{2} e n x + 3 \, {\left (b^{2} e \log \relax (c) + a b e\right )} x + 3 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x^{\frac {1}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right ) + 3 \, {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x + 3 \, {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x^{\frac {1}{3}}}{3 \, {\left (e x^{4} + d x^{\frac {10}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^3} \,d x \]
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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