Optimal. Leaf size=591 \[ -\frac {\text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e}-\frac {\text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{e}+\frac {\text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{e}+\frac {\log (d+e x) \text {Li}_2(c (a+b x))}{e}+\frac {\text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )}{e}+\frac {\text {Li}_2(1-c (a+b x)) \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )\right )}{e}+\frac {\left (\log \left (\frac {-a c e+b c d+e}{b c (d+e x)}\right )-\log \left (\frac {(a+b x) (-a c e+b c d+e)}{b (d+e x)}\right )+\log (c (a+b x))\right ) \log ^2\left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{2 e}-\frac {\left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac {\log (d+e x) \log (c (a+b x)) \log (1-c (a+b x))}{e}-\frac {\text {Li}_3(1-c (a+b x))}{e}-\frac {\text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e} \]
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Rubi [A] time = 0.52, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6597, 2440, 2435} \[ -\frac {\text {PolyLog}\left (3,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\text {PolyLog}\left (3,\frac {(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}-\frac {\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text {PolyLog}\left (2,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text {PolyLog}\left (2,\frac {(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}+\frac {\log (d+e x) \text {PolyLog}(2,c (a+b x))}{e}+\frac {\text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )}{e}+\frac {\text {PolyLog}(2,1-c (a+b x)) \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )\right )}{e}-\frac {\text {PolyLog}(3,1-c (a+b x))}{e}-\frac {\text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {\left (\log \left (\frac {-a c e+b c d+e}{b c (d+e x)}\right )-\log \left (\frac {(a+b x) (-a c e+b c d+e)}{b (d+e x)}\right )+\log (c (a+b x))\right ) \log ^2\left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{2 e}-\frac {\left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac {\log (d+e x) \log (c (a+b x)) \log (1-c (a+b x))}{e} \]
Antiderivative was successfully verified.
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Rule 2435
Rule 2440
Rule 6597
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx &=\frac {\log (d+e x) \text {Li}_2(c (a+b x))}{e}+\frac {b \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{e}\\ &=\frac {\log (d+e x) \text {Li}_2(c (a+b x))}{e}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{e}\\ &=\frac {\left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 e}+\frac {\log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{e}-\frac {\left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac {\log (d+e x) \text {Li}_2(c (a+b x))}{e}+\frac {\left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {\left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{e}-\frac {\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e}-\frac {\text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {\text {Li}_3(1-c (a+b x))}{e}-\frac {\text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 622, normalized size = 1.05 \[ \frac {-\text {Li}_3\left (\frac {b c (d+e x)}{e (a c+b x c-1)}\right )+\text {Li}_3\left (-\frac {b (d+e x)}{(b d-a e) (a c+b x c-1)}\right )+\left (\text {Li}_2\left (\frac {b c (d+e x)}{e (a c+b x c-1)}\right )-\text {Li}_2\left (-\frac {b (d+e x)}{(b d-a e) (a c+b x c-1)}\right )\right ) \log \left (-\frac {b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log (d+e x) \text {Li}_2(c (a+b x))+\text {Li}_2(-a c-b x c+1) \left (\log (d+e x)-\log \left (-\frac {b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )\right )+\text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right ) \left (\log \left (-\frac {b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log (-a c-b c x+1)\right )+\frac {1}{2} \left (-\log \left (\frac {(a+b x) (-a c e+b c d+e)}{(a c+b c x-1) (b d-a e)}\right )+\log \left (\frac {-a c e+b c d+e}{-a c e-b c e x+e}\right )+\log (c (a+b x))\right ) \log ^2\left (-\frac {b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac {e (a+b x)}{a e-b d}\right )-\log (c (a+b x))\right ) \log \left (-\frac {b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log (d+e x) \log (c (a+b x)) \log (-a c-b c x+1)+\frac {1}{2} \log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (\log (c (a+b x))-\log \left (\frac {e (a+b x)}{a e-b d}\right )\right ) \left (\log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \log (-a c-b c x+1)\right )-\text {Li}_3(-a c-b x c+1)-\text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{e x + d}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (2, c \left (b x +a \right )\right )}{e x +d}\, dx \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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