Optimal. Leaf size=40 \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2430, 6724}
\begin {gather*} \frac {\text {PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{q}-\frac {b n \text {PolyLog}\left (k+2,e x^q\right )}{q^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2430
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_k\left (e x^q\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {(b n) \int \frac {\text {Li}_{1+k}\left (e x^q\right )}{x} \, dx}{q}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.28 \begin {gather*} \frac {a \text {Li}_{1+k}\left (e x^q\right )}{q}+\frac {b \log \left (c x^n\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (k , e \,x^{q}\right )}{x}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{k}\left (e x^{q}\right )}{x}\, dx \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 = -0.02 \begin {gather*} \int \frac {\mathrm {polylog}\left (k,e\,x^q\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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