Optimal. Leaf size=70 \[ -2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )+8 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-16 b^2 n^2 \text {Li}_4\left (-d f \sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2421, 2430,
6724} \begin {gather*} 8 b n \text {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-16 b^2 n^2 \text {PolyLog}\left (4,-d f \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2421
Rule 2430
Rule 6724
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx &=-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )+(4 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )+8 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-\left (8 b^2 n^2\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )+8 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-16 b^2 n^2 \text {Li}_4\left (-d f \sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 70, normalized size = 1.00 \begin {gather*} -2 \left (\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b n \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )\right )+2 b n \text {Li}_4\left (-d f \sqrt {x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x}\, dx\]
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 = -0.01 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________