Optimal. Leaf size=448 \[ \frac {b^3 c^2 \log (-a c-b c x+1)}{6 e (b d-a e) (-a c e+b c d+e)^2}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (-a c e+b c d+e)^2}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{3 e (b d-a e)^3}+\frac {b^3 c \log (-a c-b c x+1)}{3 e (b d-a e)^2 (-a c e+b c d+e)}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (-a c e+b c d+e)}+\frac {b^3 \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}+\frac {b^2 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}-\frac {b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \]
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Rubi [A] time = 0.43, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6598, 2418, 2393, 2391, 2395, 44, 36, 31, 2394} \[ \frac {b^3 \text {PolyLog}(2,c (a+b x))}{3 e (b d-a e)^3}+\frac {b^3 \text {PolyLog}\left (2,\frac {e (-a c-b c x+1)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac {\text {PolyLog}(2,c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 c^2 \log (-a c-b c x+1)}{6 e (b d-a e) (-a c e+b c d+e)^2}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (-a c e+b c d+e)^2}+\frac {b^2 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}+\frac {b^3 c \log (-a c-b c x+1)}{3 e (b d-a e)^2 (-a c e+b c d+e)}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (-a c e+b c d+e)}+\frac {b^3 \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}-\frac {b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac {b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 44
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2418
Rule 6598
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{(d+e x)^4} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \int \frac {\log (1-a c-b c x)}{(a+b x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \int \left (\frac {b^3 \log (1-a c-b c x)}{(b d-a e)^3 (a+b x)}-\frac {e \log (1-a c-b c x)}{(b d-a e) (d+e x)^3}-\frac {b e \log (1-a c-b c x)}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e \log (1-a c-b c x)}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{3 (b d-a e)^3}-\frac {b^4 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 e (b d-a e)^3}+\frac {b^2 \int \frac {\log (1-a c-b c x)}{(d+e x)^2} \, dx}{3 (b d-a e)^2}+\frac {b \int \frac {\log (1-a c-b c x)}{(d+e x)^3} \, dx}{3 (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 e (b d-a e)^3}+\frac {\left (b^4 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}{3 e (b d-a e)^3}-\frac {\left (b^3 c\right ) \int \frac {1}{(1-a c-b c x) (d+e x)} \, dx}{3 e (b d-a e)^2}-\frac {\left (b^2 c\right ) \int \frac {1}{(1-a c-b c x) (d+e x)^2} \, dx}{6 e (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b^3 \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 )}{3 e (b d-a e)^3}-\frac {\left (b^2 c\right ) \int \left (-\frac {b^2 c^2}{(b c d+e-a c e)^2 (-1+a c+b c x)}+\frac {e}{(b c d+(1-a c) e) (d+e x)^2}+\frac {b c e}{(b c d+(1-a c) e)^2 (d+e x)}\right ) \, dx}{6 e (b d-a e)}-\frac {\left (b^3 c\right ) \int \frac {1}{d+e x} \, dx}{3 (b d-a e)^2 (b c d+e-a c e)}-\frac {\left (b^4 c^2\right ) \int \frac {1}{1-a c-b c x} \, dx}{3 e (b d-a e)^2 (b c d+e-a c e)}\\ &=\frac {b^2 c}{6 e (b d-a e) (b c d+e-a c e) (d+e x)}+\frac {b^3 c^2 \log (1-a c-b c x)}{6 e (b d-a e) (b c d+e-a c e)^2}+\frac {b^3 c \log (1-a c-b c x)}{3 e (b d-a e)^2 (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (b c d+e-a c e)^2}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (b c d+e-a c e)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 313, normalized size = 0.70 \[ \frac {\frac {b \left (2 b^2 \text {Li}_2\left (\frac {e (a c+b x c-1)}{(a c-1) e-b c d}\right )+\frac {2 b^2 c (b d-a e) (\log (-a c-b c x+1)-\log (d+e x))}{-a c e+b c d+e}+2 b^2 \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )+2 b^2 \text {Li}_2(c (a+b x))-\frac {2 b (b d-a e) \log (-a c-b c x+1)}{d+e x}+\frac {b c (b d-a e)^2 (b c (d+e x) \log (-a c-b c x+1)-a c e-b c (d+e x) \log (d+e x)+b c d+e)}{(d+e x) (-a c e+b c d+e)^2}-\frac {(b d-a e)^2 \log (-a c-b c x+1)}{(d+e x)^2}\right )}{(b d-a e)^3}-\frac {2 \text {Li}_2(c (a+b x))}{(d+e x)^3}}{6 e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1075, normalized size = 2.40 \[ -\frac {b^{3} c^{3} \polylog \left (2, b c x +a c \right )}{3 \left (b c e x +b c d \right )^{3} e}-\frac {b^{3} \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 )}{3 e \left (a e -b d \right )^{3}}-\frac {b^{3} \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 )}{3 e \left (a e -b d \right )^{3}}-\frac {b^{3} \dilog \left (-b c x -a c +1\right )}{3 e \left (a e -b d \right )^{3}}+\frac {b^{3} c \ln \left (a c e -b c d +e \left (-b c x -a c +1\right )-e \right )}{3 e \left (a e -b d \right )^{2} \left (a c e -b c d -e \right )}+\frac {b^{4} c^{2} \ln \left (-b c x -a c +1\right ) x}{3 \left (a e -b d \right )^{2} \left (a c e -b c d -e \right ) \left (-b c e x -b c d \right )}+\frac {b^{3} c^{2} \ln \left (-b c x -a c +1\right ) a}{3 \left (a e -b d \right )^{2} \left (a c e -b c d -e \right ) \left (-b c e x -b c d \right )}-\frac {b^{3} c \ln \left (-b c x -a c +1\right )}{3 \left (a e -b d \right )^{2} \left (a c e -b c d -e \right ) \left (-b c e x -b c d \right )}+\frac {b^{3} c^{2} \ln \left (a c e -b c d +e \left (-b c x -a c +1\right )-e \right )}{6 e \left (a e -b d \right ) \left (a c e -b c d -e \right )^{2}}-\frac {b^{3} c^{3} a}{6 \left (a e -b d \right ) \left (a c e -b c d -e \right )^{2} \left (-b c e x -b c d \right )}+\frac {b^{4} c^{3} d}{6 e \left (a e -b d \right ) \left (a c e -b c d -e \right )^{2} \left (-b c e x -b c d \right )}+\frac {b^{3} c^{2}}{6 \left (a e -b d \right ) \left (a c e -b c d -e \right )^{2} \left (-b c e x -b c d \right )}-\frac {b^{5} c^{4} e \ln \left (-b c x -a c +1\right ) x^{2}}{6 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}-\frac {b^{5} c^{4} \ln \left (-b c x -a c +1\right ) x d}{3 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}+\frac {b^{3} c^{4} e \ln \left (-b c x -a c +1\right ) a^{2}}{6 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}-\frac {b^{4} c^{4} \ln \left (-b c x -a c +1\right ) a d}{3 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}-\frac {b^{3} c^{3} e \ln \left (-b c x -a c +1\right ) a}{3 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}+\frac {b^{4} c^{3} \ln \left (-b c x -a c +1\right ) d}{3 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}}+\frac {b^{3} c^{2} e \ln \left (-b c x -a c +1\right )}{6 \left (a e -b d \right ) \left (-b c e x -b c d \right )^{2} \left (a c e -b c d -e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 1428, normalized size = 3.19 \[ \text {result too large to display} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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