Optimal. Leaf size=117 \[ \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+4 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2422, 2375,
2421, 6724} \begin {gather*} -2 k \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b k n \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 b n}-\frac {k \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2375
Rule 2421
Rule 2422
Rule 6724
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {(f k) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{4 b n}\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+k \int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+(2 b k n) \int \frac {\text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+4 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 186, normalized size = 1.59 \begin {gather*} \frac {1}{2} \left (4 a \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )-b n \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log ^2(x)+b k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+2 b \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log (x) \log \left (c x^n\right )-2 b k \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )+4 a k \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )-4 b k \log \left (c x^n\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+8 b k n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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