Optimal. Leaf size=177 \[ \frac{(d x)^{m+1} \text{PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}+\frac{q \text{Unintegrable}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{m+1}-\frac{b n (d x)^{m+1} \text{PolyLog}\left (2,e x^q\right )}{d (m+1)^2}-\frac{b e n q^2 x^{q+1} (d x)^m \, _2F_1\left (1,\frac{m+q+1}{q};\frac{m+2 q+1}{q};e x^q\right )}{(m+1)^3 (m+q+1)}-\frac{b n q (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^3} \]
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Rubi [A] time = 0.103605, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right ) \, dx &=-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{(b n q) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ &=-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{\left (b e n q^2\right ) \int \frac{x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^3}\\ &=-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac{\left (b e n q^2 x^{-m} (d x)^m\right ) \int \frac{x^{m+q}}{1-e x^q} \, dx}{(1+m)^3}\\ &=-\frac{b e n q^2 x^{1+q} (d x)^m \, _2F_1\left (1,\frac{1+m+q}{q};\frac{1+m+2 q}{q};e x^q\right )}{(1+m)^3 (1+m+q)}-\frac{b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac{b n (d x)^{1+m} \text{Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (e x^q\right )}{d (1+m)}+\frac{q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}\\ \end{align*}
Mathematica [A] time = 0.107549, size = 0, normalized size = 0. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.267, size = 867, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (b d^{m} m^{2} + 2 \, b d^{m} m + b d^{m}\right )} x x^{m} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} m^{2} + 2 \,{\left (b \log \left (c\right ) + a\right )} d^{m} m +{\left (b \log \left (c\right ) + a\right )} d^{m} -{\left (b d^{m} m + b d^{m}\right )} n\right )} x x^{m}\right )}{\rm Li}_2\left (e x^{q}\right ) +{\left ({\left (b d^{m} m + b d^{m}\right )} q x x^{m} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} m - 2 \, b d^{m} n +{\left (b \log \left (c\right ) + a\right )} d^{m}\right )} q x x^{m}\right )} \log \left (-e x^{q} + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \int -\frac{{\left (b d^{m} e m + b d^{m} e\right )} q^{2} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )} \log \left (x^{n}\right ) +{\left ({\left (b \log \left (c\right ) + a\right )} d^{m} e m - 2 \, b d^{m} e n +{\left (b \log \left (c\right ) + a\right )} d^{m} e\right )} q^{2} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} -{\left (e m^{3} + 3 \, e m^{2} + 3 \, e m + e\right )} x^{q} + 3 \, m + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m} b{\rm Li}_2\left (e x^{q}\right ) \log \left (c x^{n}\right ) + \left (d x\right )^{m} a{\rm Li}_2\left (e x^{q}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m}{\rm Li}_2\left (e x^{q}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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