Optimal. Leaf size=260 \[ \frac {a^3 \text {Li}_2(c (a+b x))}{3 b^3}-\frac {a^2 (-a c-b c x+1) \log (-a c-b c x+1)}{3 b^3 c}-\frac {a^2 x}{3 b^2}-\frac {(1-a c)^3 \log (-a c-b c x+1)}{9 b^3 c^3}+\frac {a (1-a c)^2 \log (-a c-b c x+1)}{6 b^3 c^2}-\frac {x (1-a c)^2}{9 b^2 c^2}+\frac {a x (1-a c)}{6 b^2 c}+\frac {1}{3} x^3 \text {Li}_2(c (a+b x))+\frac {1}{9} x^3 \log (-a c-b c x+1)-\frac {x^2 (1-a c)}{18 b c}-\frac {a x^2 \log (-a c-b c x+1)}{6 b}+\frac {a x^2}{12 b}-\frac {x^3}{27} \]
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Rubi [A] time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {6598, 43, 2416, 2389, 2295, 2395, 2393, 2391} \[ \frac {a^3 \text {PolyLog}(2,c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \text {PolyLog}(2,c (a+b x))-\frac {a^2 (-a c-b c x+1) \log (-a c-b c x+1)}{3 b^3 c}-\frac {a^2 x}{3 b^2}-\frac {x (1-a c)^2}{9 b^2 c^2}+\frac {a (1-a c)^2 \log (-a c-b c x+1)}{6 b^3 c^2}-\frac {(1-a c)^3 \log (-a c-b c x+1)}{9 b^3 c^3}+\frac {a x (1-a c)}{6 b^2 c}-\frac {x^2 (1-a c)}{18 b c}-\frac {a x^2 \log (-a c-b c x+1)}{6 b}+\frac {1}{9} x^3 \log (-a c-b c x+1)+\frac {a x^2}{12 b}-\frac {x^3}{27} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2395
Rule 2416
Rule 6598
Rubi steps
\begin {align*} \int x^2 \text {Li}_2(c (a+b x)) \, dx &=\frac {1}{3} x^3 \text {Li}_2(c (a+b x))+\frac {1}{3} b \int \frac {x^3 \log (1-a c-b c x)}{a+b x} \, dx\\ &=\frac {1}{3} x^3 \text {Li}_2(c (a+b x))+\frac {1}{3} b \int \left (\frac {a^2 \log (1-a c-b c x)}{b^3}-\frac {a x \log (1-a c-b c x)}{b^2}+\frac {x^2 \log (1-a c-b c x)}{b}-\frac {a^3 \log (1-a c-b c x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {1}{3} x^3 \text {Li}_2(c (a+b x))+\frac {1}{3} \int x^2 \log (1-a c-b c x) \, dx+\frac {a^2 \int \log (1-a c-b c x) \, dx}{3 b^2}-\frac {a^3 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 b^2}-\frac {a \int x \log (1-a c-b c x) \, dx}{3 b}\\ &=-\frac {a x^2 \log (1-a c-b c x)}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x)+\frac {1}{3} x^3 \text {Li}_2(c (a+b x))-\frac {a^3 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 b^3}-\frac {a^2 \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}-\frac {1}{6} (a c) \int \frac {x^2}{1-a c-b c x} \, dx+\frac {1}{9} (b c) \int \frac {x^3}{1-a c-b c x} \, dx\\ &=-\frac {a^2 x}{3 b^2}-\frac {a x^2 \log (1-a c-b c x)}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x)-\frac {a^2 (1-a c-b c x) \log (1-a c-b c x)}{3 b^3 c}+\frac {a^3 \text {Li}_2(c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \text {Li}_2(c (a+b x))-\frac {1}{6} (a 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 {1}{9} (b c) \int \left (-\frac {(-1+a c)^2}{b^3 c^3}+\frac {(-1+a c) x}{b^2 c^2}-\frac {x^2}{b c}+\frac {(-1+a c)^3}{b^3 c^3 (-1+a c+b c x)}\right ) \, dx\\ &=-\frac {a^2 x}{3 b^2}+\frac {a (1-a c) x}{6 b^2 c}-\frac {(1-a c)^2 x}{9 b^2 c^2}+\frac {a x^2}{12 b}-\frac {(1-a c) x^2}{18 b c}-\frac {x^3}{27}+\frac {a (1-a c)^2 \log (1-a c-b c x)}{6 b^3 c^2}-\frac {(1-a c)^3 \log (1-a c-b c x)}{9 b^3 c^3}-\frac {a x^2 \log (1-a c-b c x)}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x)-\frac {a^2 (1-a c-b c x) \log (1-a c-b c x)}{3 b^3 c}+\frac {a^3 \text {Li}_2(c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \text {Li}_2(c (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.20, size = 144, normalized size = 0.55 \[ \frac {36 c^3 \left (a^3+b^3 x^3\right ) \text {Li}_2(c (a+b x))-b c x \left (66 a^2 c^2-3 a c (5 b c x+14)+4 b^2 c^2 x^2+6 b c x+12\right )+6 \left (11 a^3 c^3+6 a^2 c^2 (b c x-3)+a \left (9 c-3 b^2 c^3 x^2\right )+2 b^3 c^3 x^3-2\right ) \log (-a c-b c x+1)}{108 b^3 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 165, normalized size = 0.63 \[ -\frac {4 \, b^{3} c^{3} x^{3} - 3 \, {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} x^{2} + 6 \, {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} x - 36 \, {\left (b^{3} c^{3} x^{3} + a^{3} c^{3}\right )} {\rm Li}_2\left (b c x + a c\right ) - 6 \, {\left (2 \, b^{3} c^{3} x^{3} - 3 \, a b^{2} c^{3} x^{2} + 6 \, a^{2} b c^{3} x + 11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
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 \[ \int x^{2} {\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 269, normalized size = 1.03 \[ \frac {11}{54 b^{3} c^{3}}-\frac {\ln \left (-b c x -a c +1\right )}{9 b^{3} c^{3}}-\frac {85 a^{3}}{108 b^{3}}+\frac {13 a^{2}}{9 b^{3} c}-\frac {x}{9 b^{2} c^{2}}-\frac {31 a}{36 b^{3} c^{2}}+\frac {\ln \left (-b c x -a c +1\right ) x \,a^{2}}{3 b^{2}}-\frac {a \,x^{2} \ln \left (-b c x -a c +1\right )}{6 b}+\frac {7 x a}{18 b^{2} c}+\frac {5 a \,x^{2}}{36 b}+\frac {\polylog \left (2, b c x +a c \right ) x^{3}}{3}+\frac {x^{3} \ln \left (-b c x -a c +1\right )}{9}-\frac {11 a^{2} x}{18 b^{2}}+\frac {11 \ln \left (-b c x -a c +1\right ) a^{3}}{18 b^{3}}-\frac {\ln \left (-b c x -a c +1\right ) a^{2}}{b^{3} c}-\frac {x^{2}}{18 b c}+\frac {\ln \left (-b c x -a c +1\right ) a}{2 b^{3} c^{2}}-\frac {x^{3}}{27}+\frac {\dilog \left (-b c x -a c +1\right ) a^{3}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 200, normalized size = 0.77 \[ -\frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} a^{3}}{3 \, b^{3}} + \frac {36 \, b^{3} c^{3} x^{3} {\rm Li}_2\left (b c x + a c\right ) - 4 \, b^{3} c^{3} x^{3} + 3 \, {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} x^{2} - 6 \, {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} x + 6 \, {\left (2 \, b^{3} c^{3} x^{3} - 3 \, a b^{2} c^{3} x^{2} + 6 \, a^{2} b c^{3} x + 11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
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 \[ \int x^2\,\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.80, size = 236, normalized size = 0.91 \[ \begin {cases} 0 & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\\frac {x^{3} \operatorname {Li}_{2}\left (a c\right )}{3} & \text {for}\: b = 0 \\- \frac {11 a^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{18 b^{3}} + \frac {a^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{3 b^{3}} - \frac {a^{2} x \operatorname {Li}_{1}\left (a c + b c x\right )}{3 b^{2}} - \frac {11 a^{2} x}{18 b^{2}} + \frac {a^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{3} c} + \frac {a x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{6 b} + \frac {5 a x^{2}}{36 b} + \frac {7 a x}{18 b^{2} c} - \frac {a \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{3} c^{2}} - \frac {x^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{9} + \frac {x^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{3} - \frac {x^{3}}{27} - \frac {x^{2}}{18 b c} - \frac {x}{9 b^{2} c^{2}} + \frac {\operatorname {Li}_{1}\left (a c + b c x\right )}{9 b^{3} c^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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