Optimal. Leaf size=198 \[ -\frac {3 a x}{4 b}+\frac {(1-a c) x}{8 b c}+\frac {x^2}{16}+\frac {(1-a c)^2 \log (1-a c-b c x)}{8 b^2 c^2}-\frac {1}{8} x^2 \log (1-a c-b c x)-\frac {3 a (1-a c-b c x) \log (1-a c-b c x)}{4 b^2 c}+\frac {3 a^2 \text {PolyLog}(2,c (a+b x))}{4 b^2}+\frac {a x \text {PolyLog}(2,c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {PolyLog}(2,c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {PolyLog}(3,c (a+b x))}{2 b^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 12, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {6734, 6730,
2494, 2436, 2332, 2468, 2440, 2438, 6733, 45, 2463, 2442} \begin {gather*} -\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}+\frac {3 a^2 \text {Li}_2(c (a+b x))}{4 b^2}+\frac {(1-a c)^2 \log (-a c-b c x+1)}{8 b^2 c^2}-\frac {3 a (-a c-b c x+1) \log (-a c-b c x+1)}{4 b^2 c}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))+\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{8} x^2 \log (-a c-b c x+1)+\frac {x (1-a c)}{8 b c}-\frac {3 a x}{4 b}+\frac {x^2}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2442
Rule 2463
Rule 2468
Rule 2494
Rule 6730
Rule 6733
Rule 6734
Rubi steps
\begin {align*} \int x \text {Li}_3(c (a+b x)) \, dx &=-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}+\frac {\int (a \text {Li}_2(c (a+b x))-b x \text {Li}_2(c (a+b x))) \, dx}{2 b}\\ &=-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}-\frac {1}{2} \int x \text {Li}_2(c (a+b x)) \, dx+\frac {a \int \text {Li}_2(c (a+b x)) \, dx}{2 b}\\ &=\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}+\frac {a \int \log (1-c (a+b x)) \, dx}{2 b}-\frac {a^2 \int \frac {\log (1-c (a+b x))}{a+b x} \, dx}{2 b}-\frac {1}{4} b \int \frac {x^2 \log (1-a c-b c x)}{a+b x} \, dx\\ &=\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}+\frac {a \int \log (1-a c-b c x) \, dx}{2 b}-\frac {a^2 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{2 b}-\frac {1}{4} b \int \left (-\frac {a \log (1-a c-b c x)}{b^2}+\frac {x \log (1-a c-b c x)}{b}+\frac {a^2 \log (1-a c-b c x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}-\frac {1}{4} \int x \log (1-a c-b c x) \, dx-\frac {a^2 \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 b^2}+\frac {a \int \log (1-a c-b c x) \, dx}{4 b}-\frac {a^2 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{4 b}-\frac {a \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{2 b^2 c}\\ &=-\frac {a x}{2 b}-\frac {1}{8} x^2 \log (1-a c-b c x)-\frac {a (1-a c-b c x) \log (1-a c-b c x)}{2 b^2 c}+\frac {a^2 \text {Li}_2(c (a+b x))}{2 b^2}+\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}-\frac {a^2 \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{4 b^2}-\frac {a \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{4 b^2 c}-\frac {1}{8} (b c) \int \frac {x^2}{1-a c-b c x} \, dx\\ &=-\frac {3 a x}{4 b}-\frac {1}{8} x^2 \log (1-a c-b c x)-\frac {3 a (1-a c-b c x) \log (1-a c-b c x)}{4 b^2 c}+\frac {3 a^2 \text {Li}_2(c (a+b x))}{4 b^2}+\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}-\frac {1}{8} (b c) \int \left (\frac {-1+a c}{b^2 c^2}-\frac {x}{b c}-\frac {(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx\\ &=-\frac {3 a x}{4 b}+\frac {(1-a c) x}{8 b c}+\frac {x^2}{16}+\frac {(1-a c)^2 \log (1-a c-b c x)}{8 b^2 c^2}-\frac {1}{8} x^2 \log (1-a c-b c x)-\frac {3 a (1-a c-b c x) \log (1-a c-b c x)}{4 b^2 c}+\frac {3 a^2 \text {Li}_2(c (a+b x))}{4 b^2}+\frac {a x \text {Li}_2(c (a+b x))}{2 b}-\frac {1}{4} x^2 \text {Li}_2(c (a+b x))-\frac {\left (a^2-b^2 x^2\right ) \text {Li}_3(c (a+b x))}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 198, normalized size = 1.00 \begin {gather*} \frac {2 a c-15 a^2 c^2+2 b c x-14 a b c^2 x+b^2 c^2 x^2+2 \log (1-a c-b c x)-16 a c \log (1-a c-b c x)+14 a^2 c^2 \log (1-a c-b c x)+12 a b c^2 x \log (1-a c-b c x)-2 b^2 c^2 x^2 \log (1-a c-b c x)+4 c^2 \left (3 a^2+2 a b x-b^2 x^2\right ) \text {PolyLog}(2,c (a+b x))-8 c^2 \left (a^2-b^2 x^2\right ) \text {PolyLog}(3,c (a+b x))}{16 b^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \polylog \left (3, c \left (b x +a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 193, normalized size = 0.97 \begin {gather*} -\frac {3 \, {\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 )} a^{2}}{4 \, b^{2}} - \frac {a^{2} {\rm Li}_{3}(b c x + a c)}{2 \, b^{2}} + \frac {8 \, b^{2} c^{2} x^{2} {\rm Li}_{3}(b c x + a c) + b^{2} c^{2} x^{2} - 2 \, {\left (7 \, a b c^{2} - b c\right )} x - 4 \, {\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x\right )} {\rm Li}_2\left (b c x + a c\right ) - 2 \, {\left (b^{2} c^{2} x^{2} - 6 \, a b c^{2} x - 7 \, a^{2} c^{2} + 8 \, a c - 1\right )} \log \left (-b c x - a c + 1\right )}{16 \, b^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 149, normalized size = 0.75 \begin {gather*} \frac {b^{2} c^{2} x^{2} - 2 \, {\left (7 \, a b c^{2} - b c\right )} x - 4 \, {\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x - 3 \, a^{2} c^{2}\right )} {\rm Li}_2\left (b c x + a c\right ) - 2 \, {\left (b^{2} c^{2} x^{2} - 6 \, a b c^{2} x - 7 \, a^{2} c^{2} + 8 \, a c - 1\right )} \log \left (-b c x - a c + 1\right ) + 8 \, {\left (b^{2} c^{2} x^{2} - a^{2} c^{2}\right )} {\rm polylog}\left (3, b c x + a c\right )}{16 \, b^{2} c^{2}} \end {gather*}
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 \begin {gather*} \int x \operatorname {Li}_{3}\left (a c + b c x\right )\, dx \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 = -0.01 \begin {gather*} \int x\,\mathrm {polylog}\left (3,c\,\left (a+b\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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