Optimal. Leaf size=278 \[ \frac {b^2 c \log (1-a c-b c x)}{2 e (b d-a e) (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{2 e (b d-a e) (d+e x)}-\frac {b^2 c \log (d+e x)}{2 e (b d-a e) (b c d+e-a c e)}+\frac {b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}+\frac {b^2 \text {PolyLog}(2,c (a+b x))}{2 e (b d-a e)^2}-\frac {\text {PolyLog}(2,c (a+b x))}{2 e (d+e x)^2}+\frac {b^2 \text {PolyLog}\left (2,\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6733, 2465,
2440, 2438, 2442, 36, 31, 2441} \begin {gather*} \frac {b^2 \text {Li}_2(c (a+b x))}{2 e (b d-a e)^2}+\frac {b^2 \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{2 e (b d-a e)^2}+\frac {b^2 c \log (-a c-b c x+1)}{2 e (b d-a e) (-a c e+b c d+e)}-\frac {b^2 c \log (d+e x)}{2 e (b d-a e) (-a c e+b c d+e)}+\frac {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 e (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}-\frac {b \log (-a c-b c x+1)}{2 e (d+e x) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2465
Rule 6733
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{(d+e x)^3} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}-\frac {b \int \frac {\log (1-a c-b c x)}{(a+b x) (d+e x)^2} \, dx}{2 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}-\frac {b \int \left (\frac {b^2 \log (1-a c-b c x)}{(b d-a e)^2 (a+b x)}-\frac {e \log (1-a c-b c x)}{(b d-a e) (d+e x)^2}-\frac {b e \log (1-a c-b c x)}{(b d-a e)^2 (d+e x)}\right ) \, dx}{2 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}+\frac {b^2 \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{2 (b d-a e)^2}-\frac {b^3 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{2 e (b d-a e)^2}+\frac {b \int \frac {\log (1-a c-b c x)}{(d+e x)^2} \, dx}{2 (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{2 e (b d-a e) (d+e x)}+\frac {b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 e (b d-a e)^2}+\frac {\left (b^3 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}{2 e (b d-a e)^2}-\frac {\left (b^2 c\right ) \int \frac {1}{(1-a c-b c x) (d+e x)} \, dx}{2 e (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{2 e (b d-a e) (d+e x)}+\frac {b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}+\frac {b^2 \text {Li}_2(c (a+b x))}{2 e (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}-\frac {b^2 \text {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 )}{2 e (b d-a e)^2}-\frac {\left (b^2 c\right ) \int \frac {1}{d+e x} \, dx}{2 (b d-a e) (b c d+e-a c e)}-\frac {\left (b^3 c^2\right ) \int \frac {1}{1-a c-b c x} \, dx}{2 e (b d-a e) (b c d+e-a c e)}\\ &=\frac {b^2 c \log (1-a c-b c x)}{2 e (b d-a e) (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{2 e (b d-a e) (d+e x)}-\frac {b^2 c \log (d+e x)}{2 e (b d-a e) (b c d+e-a c e)}+\frac {b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}+\frac {b^2 \text {Li}_2(c (a+b x))}{2 e (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{2 e (d+e x)^2}+\frac {b^2 \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 190, normalized size = 0.68 \begin {gather*} \frac {-\frac {\text {PolyLog}(2,c (a+b x))}{(d+e x)^2}+\frac {b \left (-\frac {(b d-a e) \log (1-a c-b c x)}{d+e x}+\frac {b c (b d-a e) (\log (1-a c-b c x)-\log (d+e x))}{b c d+e-a c e}+b \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )+b \text {PolyLog}(2,c (a+b x))+b \text {PolyLog}\left (2,\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )\right )}{(b d-a e)^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.12, size = 346, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {c^{3} b^{3} \polylog \left (2, x b c +a c \right )}{2 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{2} e}-\frac {c^{3} b^{3} \left (-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c \left (a e -b d \right )}\right )}{2 e}}{b c}\) | \(346\) |
default | \(\frac {-\frac {c^{3} b^{3} \polylog \left (2, x b c +a c \right )}{2 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{2} e}-\frac {c^{3} b^{3} \left (-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c \left (a e -b d \right )}\right )}{2 e}}{b c}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 482, normalized size = 1.73 \begin {gather*} -\frac {b^{2} c \log \left (x e + d\right )}{2 \, {\left (b^{2} c d^{2} e + a^{2} c e^{3} - {\left (2 \, a b c e^{2} - b e^{2}\right )} d - a e^{3}\right )}} - \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^{2}}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (\frac {b c x e + a c e - e}{b c d - a c e + e} + 1\right ) + {\rm Li}_2\left (-\frac {b c x e + a c e - e}{b c d - a c e + e}\right )\right )} b^{2}}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} - \frac {{\left (b^{2} c d^{2} + a^{2} c e^{2} - {\left (2 \, a b c e - b e\right )} d - a e^{2}\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left (b^{2} c x^{2} e^{2} + {\left (a b c e - b e\right )} d + {\left (b^{2} c d e + a b c e^{2} - b e^{2}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{2 \, {\left (b^{2} c d^{4} e - {\left (2 \, a b c e^{2} - b e^{2}\right )} d^{3} + {\left (a^{2} c e^{3} - a e^{3}\right )} d^{2} + {\left (b^{2} c d^{2} e^{3} + a^{2} c e^{5} - {\left (2 \, a b c e^{4} - b e^{4}\right )} d - a e^{5}\right )} x^{2} + 2 \, {\left (b^{2} c d^{3} e^{2} - {\left (2 \, a b c e^{3} - b e^{3}\right )} d^{2} + {\left (a^{2} c e^{4} - a e^{4}\right )} d\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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