Optimal. Leaf size=55 \[ -\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-\frac {3}{2} b n \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2441,
2352} \begin {gather*} -\frac {3}{2} b n \text {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right )-\frac {3}{2} \log \left (-\frac {e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2352
Rule 2441
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx &=-\left (\frac {3}{2} \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\frac {1}{x^{2/3}}\right )\right )\\ &=-\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{x^{2/3}}\right )\\ &=-\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-\frac {3}{2} b n \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} a \log (x)-\frac {3}{2} b \left (\log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+n \text {Li}_2\left (\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs.
\(2 (47) = 94\).
time = 0.73, size = 130, normalized size = 2.36 \begin {gather*} -\frac {3}{2} \, {\left (2 \, \log \left (d e^{\left (\frac {2}{3} \, \log \left (x\right ) - 1\right )} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-d e^{\left (\frac {2}{3} \, \log \left (x\right ) - 1\right )}\right )\right )} b n + \frac {1}{6} \, {\left (6 \, b n e \log \left (d x^{\frac {2}{3}} + e\right ) \log \left (x\right ) + 2 \, b n e \log \left (x\right )^{2} + 6 \, b d n x^{\frac {2}{3}} \log \left (x\right ) - 12 \, b e \log \left (x\right ) \log \left (x^{\frac {1}{3} \, n}\right ) - 9 \, b d n x^{\frac {2}{3}} + 6 \, {\left (b \log \left (c\right ) + a\right )} e \log \left (x\right ) - \frac {3 \, {\left (2 \, b d n x \log \left (x\right ) - 3 \, b d n x\right )}}{x^{\frac {1}{3}}}\right )} e^{\left (-1\right )} \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 = -0.02 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________