Optimal. Leaf size=55 \[ \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {3}{2} b n \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, 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 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 ) \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+e x^{2/3}\right )^n\right )}{x} \, dx &=\frac {3}{2} \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{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 ) \log \left (-\frac {e x^{2/3}}{d}\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,x^{2/3}\right )\\ &=\frac {3}{2} \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 )+\frac {3}{2} b n \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\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+e x^{2/3}\right )^n\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+n \text {Li}_2\left (\frac {d+e x^{2/3}}{d}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )}{x}\, dx\]
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. 128 vs.
\(2 (47) = 94\).
time = 0.41, size = 128, normalized size = 2.33 \begin {gather*} -\frac {3}{2} \, {\left (2 \, \log \left (x^{\frac {1}{3}}\right ) \log \left (\frac {e^{\left (\frac {2}{3} \, \log \left (x\right ) + 1\right )}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e^{\left (\frac {2}{3} \, \log \left (x\right ) + 1\right )}}{d}\right )\right )} b n + \frac {2 \, b d n \log \left (x^{\frac {2}{3}} e + d\right ) \log \left (x\right ) + 2 \, {\left (b d \log \left (c\right ) + a d\right )} \log \left (x\right ) - \frac {2 \, b n x e \log \left (x\right ) - 3 \, b n x e}{x^{\frac {1}{3}}}}{2 \, d} + \frac {3 \, {\left (2 \, b n e^{\left (\frac {2}{3} \, \log \left (x\right ) + 1\right )} \log \left (x^{\frac {1}{3}}\right ) - b n e^{\left (\frac {2}{3} \, \log \left (x\right ) + 1\right )}\right )}}{2 \, d} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}}{x}\, dx \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+e\,x^{2/3}\right )}^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________