Optimal. Leaf size=138 \[ \frac {b \text {Li}_2(c (a+b x))}{e (b d-a e)}-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}+\frac {b \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{e (b d-a e)}+\frac {b \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )}{e (b d-a e)} \]
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Rubi [A] time = 0.18, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6598, 2418, 2393, 2391, 2394} \[ \frac {b \text {PolyLog}(2,c (a+b x))}{e (b d-a e)}-\frac {\text {PolyLog}(2,c (a+b x))}{e (d+e x)}+\frac {b \text {PolyLog}\left (2,\frac {e (-a c-b c x+1)}{-a c e+b c d+e}\right )}{e (b d-a e)}+\frac {b \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )}{e (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 6598
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{(d+e x)^2} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}-\frac {b \int \frac {\log (1-a c-b c x)}{(a+b x) (d+e x)} \, dx}{e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}-\frac {b \int \left (\frac {b \log (1-a c-b c x)}{(b d-a e) (a+b x)}-\frac {e \log (1-a c-b c x)}{(b d-a e) (d+e x)}\right ) \, dx}{e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}+\frac {b \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{b d-a e}-\frac {b^2 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{e (b d-a e)}\\ &=\frac {b \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e (b d-a e)}-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{e (b d-a e)}+\frac {\left (b^2 c\right ) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{e (b d-a e)}\\ &=\frac {b \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e (b d-a e)}+\frac {b \text {Li}_2(c (a+b x))}{e (b d-a e)}-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{e (b d-a e)}\\ &=\frac {b \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e (b d-a e)}+\frac {b \text {Li}_2(c (a+b x))}{e (b d-a e)}-\frac {\text {Li}_2(c (a+b x))}{e (d+e x)}+\frac {b \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{e (b d-a e)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 108, normalized size = 0.78 \[ \frac {\frac {b \left (\text {Li}_2\left (\frac {e (a c+b x c-1)}{-b c d+a c e-e}\right )+\log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )+\text {Li}_2(c (a+b x))\right )}{b d-a e}-\frac {\text {Li}_2(c (a+b x))}{d+e x}}{e} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 189, normalized size = 1.37 \[ -\frac {b c \polylog \left (2, b c x +a c \right )}{\left (b c e x +b c d \right ) e}-\frac {b \dilog \left (\frac {a c e -b c d +e \left (-b c x -a c +1\right )-e}{a c e -b c d -e}\right )}{e \left (a e -b d \right )}-\frac {b \ln \left (-b c x -a c +1\right ) \ln \left (\frac {a c e -b c d +e \left (-b c x -a c +1\right )-e}{a c e -b c d -e}\right )}{e \left (a e -b d \right )}-\frac {b \dilog \left (-b c x -a c +1\right )}{e \left (a e -b d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 166, normalized size = 1.20 \[ -\frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b}{b d e - a e^{2}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (\frac {b c e x + {\left (a c - 1\right )} e}{b c d - {\left (a c - 1\right )} e} + 1\right ) + {\rm Li}_2\left (-\frac {b c e x + {\left (a c - 1\right )} e}{b c d - {\left (a c - 1\right )} e}\right )\right )} b}{b d e - a e^{2}} - \frac {{\rm Li}_2\left (b c x + a c\right )}{e^{2} x + d e} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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