Optimal. Leaf size=385 \[ -\frac {(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\frac {(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{6 b^3 c^2 e}-\frac {(b d-a e)^3 \text {Li}_2(c (a+b x))}{3 b^3 e}-\frac {(-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{3 b^3 c}-\frac {x (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac {x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac {x (b d-a e)^2}{3 b^2}+\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}-\frac {(d+e x)^2 (-a c e+b c d+e)}{18 b c e}+\frac {(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{6 b e}+\frac {(d+e x)^3 \log (-a c-b c x+1)}{9 e}-\frac {(d+e x)^2 (b d-a e)}{12 b e}-\frac {(d+e x)^3}{27 e} \]
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Rubi [A] time = 0.34, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6598, 2418, 2389, 2295, 2393, 2391, 2395, 43} \[ -\frac {(b d-a e)^3 \text {PolyLog}(2,c (a+b x))}{3 b^3 e}+\frac {(d+e x)^3 \text {PolyLog}(2,c (a+b x))}{3 e}-\frac {x (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac {(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{6 b^3 c^2 e}-\frac {(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\frac {x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac {(-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{3 b^3 c}-\frac {x (b d-a e)^2}{3 b^2}-\frac {(d+e x)^2 (-a c e+b c d+e)}{18 b c e}+\frac {(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{6 b e}+\frac {(d+e x)^3 \log (-a c-b c x+1)}{9 e}-\frac {(d+e x)^2 (b d-a e)}{12 b e}-\frac {(d+e x)^3}{27 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2395
Rule 2418
Rule 6598
Rubi steps
\begin {align*} \int (d+e x)^2 \text {Li}_2(c (a+b x)) \, dx &=\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}+\frac {b \int \frac {(d+e x)^3 \log (1-a c-b c x)}{a+b x} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}+\frac {b \int \left (\frac {e (b d-a e)^2 \log (1-a c-b c x)}{b^3}+\frac {(b d-a e)^3 \log (1-a c-b c x)}{b^3 (a+b x)}+\frac {e (b d-a e) (d+e x) \log (1-a c-b c x)}{b^2}+\frac {e (d+e x)^2 \log (1-a c-b c x)}{b}\right ) \, dx}{3 e}\\ &=\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}+\frac {1}{3} \int (d+e x)^2 \log (1-a c-b c x) \, dx+\frac {(b d-a e) \int (d+e x) \log (1-a c-b c x) \, dx}{3 b}+\frac {(b d-a e)^2 \int \log (1-a c-b c x) \, dx}{3 b^2}+\frac {(b d-a e)^3 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 b^2 e}\\ &=\frac {(b d-a e) (d+e x)^2 \log (1-a c-b c x)}{6 b e}+\frac {(d+e x)^3 \log (1-a c-b c x)}{9 e}+\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}+\frac {(b c) \int \frac {(d+e x)^3}{1-a c-b c x} \, dx}{9 e}+\frac {(c (b d-a e)) \int \frac {(d+e x)^2}{1-a c-b c x} \, dx}{6 e}-\frac {(b d-a e)^2 \operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}+\frac {(b d-a e)^3 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 b^3 e}\\ &=-\frac {(b d-a e)^2 x}{3 b^2}-\frac {(b d-a e)^2 (1-a c-b c x) \log (1-a c-b c x)}{3 b^3 c}+\frac {(b d-a e) (d+e x)^2 \log (1-a c-b c x)}{6 b e}+\frac {(d+e x)^3 \log (1-a c-b c x)}{9 e}-\frac {(b d-a e)^3 \text {Li}_2(c (a+b x))}{3 b^3 e}+\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}+\frac {(b c) \int \left (-\frac {e (b c d+e-a c e)^2}{b^3 c^3}+\frac {(b c d+e-a c e)^3}{b^3 c^3 (1-a c-b c x)}-\frac {e (b c d+e-a c e) (d+e x)}{b^2 c^2}-\frac {e (d+e x)^2}{b c}\right ) \, dx}{9 e}+\frac {(c (b d-a e)) \int \left (-\frac {e (b c d+e-a c e)}{b^2 c^2}+\frac {(b c d+e-a c e)^2}{b^2 c^2 (1-a c-b c x)}-\frac {e (d+e x)}{b c}\right ) \, dx}{6 e}\\ &=-\frac {(b d-a e)^2 x}{3 b^2}-\frac {(b d-a e) (b c d+e-a c e) x}{6 b^2 c}-\frac {(b c d+e-a c e)^2 x}{9 b^2 c^2}-\frac {(b d-a e) (d+e x)^2}{12 b e}-\frac {(b c d+e-a c e) (d+e x)^2}{18 b c e}-\frac {(d+e x)^3}{27 e}-\frac {(b d-a e) (b c d+e-a c e)^2 \log (1-a c-b c x)}{6 b^3 c^2 e}-\frac {(b c d+e-a c e)^3 \log (1-a c-b c x)}{9 b^3 c^3 e}-\frac {(b d-a e)^2 (1-a c-b c x) \log (1-a c-b c x)}{3 b^3 c}+\frac {(b d-a e) (d+e x)^2 \log (1-a c-b c x)}{6 b e}+\frac {(d+e x)^3 \log (1-a c-b c x)}{9 e}-\frac {(b d-a e)^3 \text {Li}_2(c (a+b x))}{3 b^3 e}+\frac {(d+e x)^3 \text {Li}_2(c (a+b x))}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 274, normalized size = 0.71 \[ \frac {b c \left (-66 a^2 c^2 e^2 x+3 a c \left (b c \left (-36 d^2+54 d e x+5 e^2 x^2\right )+14 e^2 x\right )+108 b c d^2 (a c+b c x-1) \log (1-c (a+b x))-x \left (b^2 c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+6 b c e (9 d+e x)+12 e^2\right )\right )+6 e (a c+b c x-1) \log (-a c-b c x+1) \left (e \left (11 a^2 c^2-7 a c+2\right )+b c (d (9-27 a c)+e x (2-5 a c))+b^2 c^2 x (9 d+2 e x)\right )+36 c^3 \text {Li}_2(c (a+b x)) \left (a^3 e^2-3 a^2 b d e+3 a b^2 d^2+b^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{108 b^3 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 373, normalized size = 0.97 \[ -\frac {4 \, b^{3} c^{3} e^{2} x^{3} + 3 \, {\left (9 \, b^{3} c^{3} d e - {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (18 \, b^{3} c^{3} d^{2} - 9 \, {\left (3 \, a b^{2} c^{3} - b^{2} c^{2}\right )} d e + {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} e^{2}\right )} x - 36 \, {\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x + 3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{3} d e + a^{3} c^{3} e^{2}\right )} {\rm Li}_2\left (b c x + a c\right ) - 6 \, {\left (2 \, b^{3} c^{3} e^{2} x^{3} + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} - 9 \, {\left (3 \, a^{2} b c^{3} - 4 \, a b c^{2} + b c\right )} d e + {\left (11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} e^{2} + 3 \, {\left (3 \, b^{3} c^{3} d e - a b^{2} c^{3} e^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{3} d e + a^{2} b c^{3} e^{2}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
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 \[ \int {\left (e x + d\right )}^{2} {\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 687, normalized size = 1.78 \[ -\frac {3 e \ln \left (-b c x -a c +1\right ) a^{2} d}{2 b^{2}}-\frac {e \ln \left (-b c x -a c +1\right ) d}{2 b^{2} c^{2}}-\frac {e^{2} \ln \left (-b c x -a c +1\right ) a^{2}}{b^{3} c}+\frac {7 e^{2} x a}{18 b^{2} c}-\frac {e x d}{2 b c}-\frac {e \dilog \left (-b c x -a c +1\right ) a^{2} d}{b^{2}}+\frac {d^{2}}{b c}+\frac {11 e^{2}}{54 b^{3} c^{3}}+\frac {e^{2} \ln \left (-b c x -a c +1\right ) x^{3}}{9}+\ln \left (-b c x -a c +1\right ) x \,d^{2}-\frac {e \,x^{2} d}{4}-\frac {a \,d^{2}}{b}+\frac {3 e d}{4 b^{2} c^{2}}+\frac {13 e^{2} a^{2}}{9 b^{3} c}-\frac {31 e^{2} a}{36 b^{3} c^{2}}-\frac {85 e^{2} a^{3}}{108 b^{3}}+\frac {\polylog \left (2, b c x +a c \right ) d^{3}}{3 e}+\frac {e^{2} \polylog \left (2, b c x +a c \right ) x^{3}}{3}+\polylog \left (2, b c x +a c \right ) x \,d^{2}-\frac {\dilog \left (-b c x -a c +1\right ) d^{3}}{3 e}-d^{2} x -\frac {e^{2} x^{3}}{27}-\frac {e^{2} x}{9 b^{2} c^{2}}-\frac {x^{2} e^{2}}{18 b c}-\frac {\ln \left (-b c x -a c +1\right ) d^{2}}{b c}-\frac {e^{2} \ln \left (-b c x -a c +1\right )}{9 b^{3} c^{3}}+\frac {e \ln \left (-b c x -a c +1\right ) x^{2} d}{2}+\frac {\ln \left (-b c x -a c +1\right ) a \,d^{2}}{b}+\frac {11 e^{2} \ln \left (-b c x -a c +1\right ) a^{3}}{18 b^{3}}+\frac {5 e^{2} x^{2} a}{36 b}-\frac {11 e^{2} x \,a^{2}}{18 b^{2}}+e \polylog \left (2, b c x +a c \right ) d \,x^{2}+\frac {\dilog \left (-b c x -a c +1\right ) a \,d^{2}}{b}+\frac {e^{2} \dilog \left (-b c x -a c +1\right ) a^{3}}{3 b^{3}}+\frac {e^{2} \ln \left (-b c x -a c +1\right ) a}{2 b^{3} c^{2}}+\frac {3 e x a d}{2 b}-\frac {e^{2} \ln \left (-b c x -a c +1\right ) x^{2} a}{6 b}+\frac {e^{2} \ln \left (-b c x -a c +1\right ) x \,a^{2}}{3 b^{2}}+\frac {2 e \ln \left (-b c x -a c +1\right ) a d}{b^{2} c}-\frac {e \ln \left (-b c x -a c +1\right ) x a d}{b}+\frac {7 e \,a^{2} d}{4 b^{2}}-\frac {5 e a d}{2 b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 406, normalized size = 1.05 \[ -\frac {{\left (3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2}\right )} {\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )}}{3 \, b^{3}} - \frac {4 \, b^{3} c^{3} e^{2} x^{3} + 3 \, {\left (9 \, b^{3} c^{3} d e - {\left (5 \, a b^{2} c^{3} - 2 \, b^{2} c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (18 \, b^{3} c^{3} d^{2} - 9 \, {\left (3 \, a b^{2} c^{3} - b^{2} c^{2}\right )} d e + {\left (11 \, a^{2} b c^{3} - 7 \, a b c^{2} + 2 \, b c\right )} e^{2}\right )} x - 36 \, {\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x\right )} {\rm Li}_2\left (b c x + a c\right ) - 6 \, {\left (2 \, b^{3} c^{3} e^{2} x^{3} + 18 \, {\left (a b^{2} c^{3} - b^{2} c^{2}\right )} d^{2} - 9 \, {\left (3 \, a^{2} b c^{3} - 4 \, a b c^{2} + b c\right )} d e + {\left (11 \, a^{3} c^{3} - 18 \, a^{2} c^{2} + 9 \, a c - 2\right )} e^{2} + 3 \, {\left (3 \, b^{3} c^{3} d e - a b^{2} c^{3} e^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{3} d e + a^{2} b c^{3} e^{2}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{108 \, b^{3} c^{3}} \]
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 \[ \int \mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.00, size = 561, normalized size = 1.46 \[ \begin {cases} 0 & \text {for}\: b = 0 \wedge c = 0 \\\left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \operatorname {Li}_{2}\left (a c\right ) & \text {for}\: b = 0 \\0 & \text {for}\: c = 0 \\- \frac {11 a^{3} e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{18 b^{3}} + \frac {a^{3} e^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{3 b^{3}} + \frac {3 a^{2} d e \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{2}} - \frac {a^{2} d e \operatorname {Li}_{2}\left (a c + b c x\right )}{b^{2}} - \frac {a^{2} e^{2} x \operatorname {Li}_{1}\left (a c + b c x\right )}{3 b^{2}} - \frac {11 a^{2} e^{2} x}{18 b^{2}} + \frac {a^{2} e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{3} c} - \frac {a d^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {a d^{2} \operatorname {Li}_{2}\left (a c + b c x\right )}{b} + \frac {a d e x \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {3 a d e x}{2 b} + \frac {a e^{2} x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{6 b} + \frac {5 a e^{2} x^{2}}{36 b} - \frac {2 a d e \operatorname {Li}_{1}\left (a c + b c x\right )}{b^{2} c} + \frac {7 a e^{2} x}{18 b^{2} c} - \frac {a e^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{3} c^{2}} - d^{2} x \operatorname {Li}_{1}\left (a c + b c x\right ) + d^{2} x \operatorname {Li}_{2}\left (a c + b c x\right ) - d^{2} x - \frac {d e x^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{2} + d e x^{2} \operatorname {Li}_{2}\left (a c + b c x\right ) - \frac {d e x^{2}}{4} - \frac {e^{2} x^{3} \operatorname {Li}_{1}\left (a c + b c x\right )}{9} + \frac {e^{2} x^{3} \operatorname {Li}_{2}\left (a c + b c x\right )}{3} - \frac {e^{2} x^{3}}{27} + \frac {d^{2} \operatorname {Li}_{1}\left (a c + b c x\right )}{b c} - \frac {d e x}{2 b c} - \frac {e^{2} x^{2}}{18 b c} + \frac {d e \operatorname {Li}_{1}\left (a c + b c x\right )}{2 b^{2} c^{2}} - \frac {e^{2} x}{9 b^{2} c^{2}} + \frac {e^{2} \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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