Optimal. Leaf size=238 \[ -\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}+\frac {b e^3 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e x^{2/3}}\right )}{d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445,
2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\frac {b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{d^3}+\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 \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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
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} \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 264, normalized size = 1.11 \begin {gather*} -\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\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 d e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+3 e^2 x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 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 d-3 e x^{2/3} \log \left (d+e x^{2/3}\right )+2 e x^{2/3} \log (x)\right )-6 b e^2 n x^{4/3} \left (\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 )+b n \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )\right )\right )}{d^3}}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{2}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________