Optimal. Leaf size=245 \[ -\frac {q^2 \text {Int}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{(m+1)^2}-\frac {q (d x)^{m+1} \text {Li}_2\left (e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac {(d x)^{m+1} \text {Li}_3\left (e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}+\frac {2 b e n q^3 x^{q+1} (d x)^m \, _2F_1\left (1,\frac {m+q+1}{q};\frac {m+2 q+1}{q};e x^q\right )}{(m+1)^4 (m+q+1)}+\frac {2 b n q (d x)^{m+1} \text {Li}_2\left (e x^q\right )}{d (m+1)^3}-\frac {b n (d x)^{m+1} \text {Li}_3\left (e x^q\right )}{d (m+1)^2}+\frac {2 b n q^2 (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^4} \]
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Rubi [A] time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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\begin {align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (e x^q\right ) \, dx &=-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right ) \, dx}{1+m}+\frac {(b n q) \int (d x)^m \text {Li}_2\left (e x^q\right ) \, dx}{(1+m)^2}\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \frac {\left (b n q^2\right ) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^3}\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {\left (b e n q^3\right ) \int \frac {x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^4}\right )\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {\left (b e n q^3 x^{-m} (d x)^m\right ) \int \frac {x^{m+q}}{1-e x^q} \, dx}{(1+m)^4}\right )\\ &=2 \left (\frac {b e n q^3 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)^4 (1+m+q)}+\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}\right )+\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (d x\right )^{m} b \log \left (c x^{n}\right ) + \left (d x\right )^{m} a\right )} {\rm polylog}\left (3, e x^{q}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_{3}(e x^{q})\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.28, size = 1065, normalized size = 4.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (m^{2} q + 2 \, m q + q\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) + {\left ({\left (m^{2} q + 2 \, m q + q\right )} a d^{m} + {\left ({\left (m^{2} q + 2 \, m q + q\right )} d^{m} \log \relax (c) - 2 \, {\left (m n q + n q\right )} d^{m}\right )} b\right )} x x^{m}\right )} {\rm Li}_2\left (e x^{q}\right ) + {\left ({\left (m q^{2} + q^{2}\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) + {\left ({\left (m q^{2} + q^{2}\right )} a d^{m} - {\left (3 \, d^{m} n q^{2} - {\left (m q^{2} + q^{2}\right )} d^{m} \log \relax (c)\right )} b\right )} x x^{m}\right )} \log \left (-e x^{q} + 1\right ) - {\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} b d^{m} x x^{m} \log \left (x^{n}\right ) + {\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a d^{m} + {\left ({\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} d^{m} \log \relax (c) - {\left (m^{2} n + 2 \, m n + n\right )} d^{m}\right )} b\right )} x x^{m}\right )} {\rm Li}_{3}(e x^{q})}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} + \int -\frac {{\left (m q^{3} + q^{3}\right )} b d^{m} e e^{\left (m \log \relax (x) + q \log \relax (x)\right )} \log \left (x^{n}\right ) + {\left ({\left (m q^{3} + q^{3}\right )} a d^{m} e - {\left (3 \, d^{m} e n q^{3} - {\left (m q^{3} + q^{3}\right )} d^{m} e \log \relax (c)\right )} b\right )} e^{\left (m \log \relax (x) + q \log \relax (x)\right )}}{m^{4} + 4 \, m^{3} - {\left (m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1\right )} e x^{q} + 6 \, m^{2} + 4 \, m + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (3,e\,x^q\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x^{q}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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