Optimal. Leaf size=385 \[ -\frac{(b d-a e)^3 \text{PolyLog}(2,c (a+b x))}{3 b^3 e}+\frac{(d+e x)^3 \text{PolyLog}(2,c (a+b x))}{3 e}-\frac{x (-a c e+b c d+e)^2}{9 b^2 c^2}-\frac{(b d-a e) (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{6 b^3 c^2 e}-\frac{(-a c e+b c d+e)^3 \log (-a c-b c x+1)}{9 b^3 c^3 e}-\frac{x (b d-a e) (-a c e+b c d+e)}{6 b^2 c}-\frac{(-a c-b c x+1) (b d-a e)^2 \log (-a c-b c x+1)}{3 b^3 c}-\frac{x (b d-a e)^2}{3 b^2}-\frac{(d+e x)^2 (-a c e+b c d+e)}{18 b c e}+\frac{(d+e x)^2 (b d-a e) \log (-a c-b c x+1)}{6 b e}+\frac{(d+e x)^3 \log (-a c-b c x+1)}{9 e}-\frac{(d+e x)^2 (b d-a e)}{12 b e}-\frac{(d+e x)^3}{27 e} \]
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Rubi [A] time = 0.338798, antiderivative size = 385, normalized size of antiderivative = 1., 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 6598
Rule 2418
Rule 2389
Rule 2295
Rule 2393
Rule 2391
Rule 2395
Rule 43
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*}
Mathematica [A] time = 0.182727, size = 274, normalized size = 0.71 \[ \frac{36 c^3 \left (-3 a^2 b d e+a^3 e^2+3 a b^2 d^2+b^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text{PolyLog}(2,c (a+b x))+b c \left (-66 a^2 c^2 e^2 x+3 a c \left (b c \left (-36 d^2+54 d e x+5 e^2 x^2\right )+14 e^2 x\right )+108 b c d^2 (a c+b c x-1) \log (1-c (a+b x))-x \left (b^2 c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+6 b c e (9 d+e x)+12 e^2\right )\right )+6 e (a c+b c x-1) \log (-a c-b c x+1) \left (e \left (11 a^2 c^2-7 a c+2\right )+b c (d (9-27 a c)+e x (2-5 a c))+b^2 c^2 x (9 d+2 e x)\right )}{108 b^3 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 687, normalized size = 1.8 \begin{align*} -{d}^{2}x-{\frac{{x}^{3}{e}^{2}}{27}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){a}^{2}}{c{b}^{3}}}+{\frac{3\,axde}{2\,b}}+{\frac{7\,{e}^{2}xa}{18\,{b}^{2}c}}-{\frac{e{\it dilog} \left ( -xbc-ac+1 \right ){a}^{2}d}{{b}^{2}}}-{\frac{dxe}{2\,bc}}-{\frac{3\,e\ln \left ( -xbc-ac+1 \right ){a}^{2}d}{2\,{b}^{2}}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){x}^{2}a}{6\,b}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) x{a}^{2}}{3\,{b}^{2}}}-{\frac{e\ln \left ( -xbc-ac+1 \right ) d}{2\,{b}^{2}{c}^{2}}}-{\frac{5\,aed}{2\,{b}^{2}c}}+{\frac{7\,e{a}^{2}d}{4\,{b}^{2}}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) a}{2\,{b}^{3}{c}^{2}}}+{\frac{{d}^{2}}{bc}}+{\frac{11\,{e}^{2}}{54\,{c}^{3}{b}^{3}}}+{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ){x}^{3}}{9}}-{\frac{{\it dilog} \left ( -xbc-ac+1 \right ){d}^{3}}{3\,e}}+{\frac{{\it polylog} \left ( 2,xbc+ac \right ){d}^{3}}{3\,e}}+\ln \left ( -xbc-ac+1 \right ) x{d}^{2}+{\frac{{e}^{2}{\it polylog} \left ( 2,xbc+ac \right ){x}^{3}}{3}}+{\it polylog} \left ( 2,xbc+ac \right ) x{d}^{2}-{\frac{{x}^{2}de}{4}}+2\,{\frac{e\ln \left ( -xbc-ac+1 \right ) ad}{{b}^{2}c}}-{\frac{e\ln \left ( -xbc-ac+1 \right ) xad}{b}}+{\frac{13\,{e}^{2}{a}^{2}}{9\,c{b}^{3}}}+{\frac{3\,de}{4\,{b}^{2}{c}^{2}}}-{\frac{31\,a{e}^{2}}{36\,{b}^{3}{c}^{2}}}-{\frac{85\,{e}^{2}{a}^{3}}{108\,{b}^{3}}}-{\frac{a{d}^{2}}{b}}+{\frac{5\,{e}^{2}{x}^{2}a}{36\,b}}-{\frac{11\,{e}^{2}x{a}^{2}}{18\,{b}^{2}}}-{\frac{{e}^{2}{x}^{2}}{18\,bc}}-{\frac{{e}^{2}x}{9\,{b}^{2}{c}^{2}}}+{\frac{e\ln \left ( -xbc-ac+1 \right ){x}^{2}d}{2}}+e{\it polylog} \left ( 2,xbc+ac \right ) d{x}^{2}+{\frac{{e}^{2}{\it dilog} \left ( -xbc-ac+1 \right ){a}^{3}}{3\,{b}^{3}}}+{\frac{11\,{e}^{2}\ln \left ( -xbc-ac+1 \right ){a}^{3}}{18\,{b}^{3}}}+{\frac{{\it dilog} \left ( -xbc-ac+1 \right ) a{d}^{2}}{b}}+{\frac{\ln \left ( -xbc-ac+1 \right ) a{d}^{2}}{b}}-{\frac{\ln \left ( -xbc-ac+1 \right ){d}^{2}}{bc}}-{\frac{{e}^{2}\ln \left ( -xbc-ac+1 \right ) }{9\,{c}^{3}{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03377, size = 548, normalized size = 1.42 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32108, size = 779, normalized size = 2.02 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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