Optimal. Leaf size=605 \[ -\frac{(b d-a e)^4 \text{PolyLog}(2,c (a+b x))}{4 b^4 e}+\frac{(d+e x)^4 \text{PolyLog}(2,c (a+b x))}{4 e}-\frac{x (b d-a e) (-a c e+b c d+e)^2}{12 b^3 c^2}-\frac{(d+e x)^2 (-a c e+b c d+e)^2}{32 b^2 c^2 e}-\frac{x (-a c e+b c d+e)^3}{16 b^3 c^3}-\frac{(b d-a e)^2 (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{8 b^4 c^2 e}-\frac{(b d-a e) (-a c e+b c d+e)^3 \log (-a c-b c x+1)}{12 b^4 c^3 e}-\frac{(-a c e+b c d+e)^4 \log (-a c-b c x+1)}{16 b^4 c^4 e}-\frac{x (b d-a e)^2 (-a c e+b c d+e)}{8 b^3 c}-\frac{(d+e x)^2 (b d-a e) (-a c e+b c d+e)}{24 b^2 c e}-\frac{(-a c-b c x+1) (b d-a e)^3 \log (-a c-b c x+1)}{4 b^4 c}+\frac{(d+e x)^2 (b d-a e)^2 \log (-a c-b c x+1)}{8 b^2 e}-\frac{x (b d-a e)^3}{4 b^3}-\frac{(d+e x)^2 (b d-a e)^2}{16 b^2 e}-\frac{(d+e x)^3 (-a c e+b c d+e)}{48 b c e}+\frac{(d+e x)^3 (b d-a e) \log (-a c-b c x+1)}{12 b e}+\frac{(d+e x)^4 \log (-a c-b c x+1)}{16 e}-\frac{(d+e x)^3 (b d-a e)}{36 b e}-\frac{(d+e x)^4}{64 e} \]
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Rubi [A] time = 0.587251, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{(b d-a e)^4 \text{PolyLog}(2,c (a+b x))}{4 b^4 e}+\frac{(d+e x)^4 \text{PolyLog}(2,c (a+b x))}{4 e}-\frac{x (b d-a e) (-a c e+b c d+e)^2}{12 b^3 c^2}-\frac{(d+e x)^2 (-a c e+b c d+e)^2}{32 b^2 c^2 e}-\frac{x (-a c e+b c d+e)^3}{16 b^3 c^3}-\frac{(b d-a e)^2 (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{8 b^4 c^2 e}-\frac{(b d-a e) (-a c e+b c d+e)^3 \log (-a c-b c x+1)}{12 b^4 c^3 e}-\frac{(-a c e+b c d+e)^4 \log (-a c-b c x+1)}{16 b^4 c^4 e}-\frac{x (b d-a e)^2 (-a c e+b c d+e)}{8 b^3 c}-\frac{(d+e x)^2 (b d-a e) (-a c e+b c d+e)}{24 b^2 c e}-\frac{(-a c-b c x+1) (b d-a e)^3 \log (-a c-b c x+1)}{4 b^4 c}+\frac{(d+e x)^2 (b d-a e)^2 \log (-a c-b c x+1)}{8 b^2 e}-\frac{x (b d-a e)^3}{4 b^3}-\frac{(d+e x)^2 (b d-a e)^2}{16 b^2 e}-\frac{(d+e x)^3 (-a c e+b c d+e)}{48 b c e}+\frac{(d+e x)^3 (b d-a e) \log (-a c-b c x+1)}{12 b e}+\frac{(d+e x)^4 \log (-a c-b c x+1)}{16 e}-\frac{(d+e x)^3 (b d-a e)}{36 b e}-\frac{(d+e x)^4}{64 e} \]
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)^3 \text{Li}_2(c (a+b x)) \, dx &=\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}+\frac{b \int \frac{(d+e x)^4 \log (1-a c-b c x)}{a+b x} \, dx}{4 e}\\ &=\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}+\frac{b \int \left (\frac{e (b d-a e)^3 \log (1-a c-b c x)}{b^4}+\frac{(b d-a e)^4 \log (1-a c-b c x)}{b^4 (a+b x)}+\frac{e (b d-a e)^2 (d+e x) \log (1-a c-b c x)}{b^3}+\frac{e (b d-a e) (d+e x)^2 \log (1-a c-b c x)}{b^2}+\frac{e (d+e x)^3 \log (1-a c-b c x)}{b}\right ) \, dx}{4 e}\\ &=\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}+\frac{1}{4} \int (d+e x)^3 \log (1-a c-b c x) \, dx+\frac{(b d-a e) \int (d+e x)^2 \log (1-a c-b c x) \, dx}{4 b}+\frac{(b d-a e)^2 \int (d+e x) \log (1-a c-b c x) \, dx}{4 b^2}+\frac{(b d-a e)^3 \int \log (1-a c-b c x) \, dx}{4 b^3}+\frac{(b d-a e)^4 \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{4 b^3 e}\\ &=\frac{(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac{(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac{(d+e x)^4 \log (1-a c-b c x)}{16 e}+\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}+\frac{(b c) \int \frac{(d+e x)^4}{1-a c-b c x} \, dx}{16 e}+\frac{(c (b d-a e)) \int \frac{(d+e x)^3}{1-a c-b c x} \, dx}{12 e}+\frac{\left (c (b d-a e)^2\right ) \int \frac{(d+e x)^2}{1-a c-b c x} \, dx}{8 b e}-\frac{(b d-a e)^3 \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{4 b^4 c}+\frac{(b d-a e)^4 \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{4 b^4 e}\\ &=-\frac{(b d-a e)^3 x}{4 b^3}-\frac{(b d-a e)^3 (1-a c-b c x) \log (1-a c-b c x)}{4 b^4 c}+\frac{(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac{(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac{(d+e x)^4 \log (1-a c-b c x)}{16 e}-\frac{(b d-a e)^4 \text{Li}_2(c (a+b x))}{4 b^4 e}+\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}+\frac{(b c) \int \left (-\frac{e (b c d+e-a c e)^3}{b^4 c^4}+\frac{(b c d+e-a c e)^4}{b^4 c^4 (1-a c-b c x)}-\frac{e (b c d+e-a c e)^2 (d+e x)}{b^3 c^3}-\frac{e (b c d+e-a c e) (d+e x)^2}{b^2 c^2}-\frac{e (d+e x)^3}{b c}\right ) \, dx}{16 e}+\frac{(c (b d-a e)) \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}{12 e}+\frac{\left (c (b d-a e)^2\right ) \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}{8 b e}\\ &=-\frac{(b d-a e)^3 x}{4 b^3}-\frac{(b d-a e)^2 (b c d+e-a c e) x}{8 b^3 c}-\frac{(b d-a e) (b c d+e-a c e)^2 x}{12 b^3 c^2}-\frac{(b c d+e-a c e)^3 x}{16 b^3 c^3}-\frac{(b d-a e)^2 (d+e x)^2}{16 b^2 e}-\frac{(b d-a e) (b c d+e-a c e) (d+e x)^2}{24 b^2 c e}-\frac{(b c d+e-a c e)^2 (d+e x)^2}{32 b^2 c^2 e}-\frac{(b d-a e) (d+e x)^3}{36 b e}-\frac{(b c d+e-a c e) (d+e x)^3}{48 b c e}-\frac{(d+e x)^4}{64 e}-\frac{(b d-a e)^2 (b c d+e-a c e)^2 \log (1-a c-b c x)}{8 b^4 c^2 e}-\frac{(b d-a e) (b c d+e-a c e)^3 \log (1-a c-b c x)}{12 b^4 c^3 e}-\frac{(b c d+e-a c e)^4 \log (1-a c-b c x)}{16 b^4 c^4 e}-\frac{(b d-a e)^3 (1-a c-b c x) \log (1-a c-b c x)}{4 b^4 c}+\frac{(b d-a e)^2 (d+e x)^2 \log (1-a c-b c x)}{8 b^2 e}+\frac{(b d-a e) (d+e x)^3 \log (1-a c-b c x)}{12 b e}+\frac{(d+e x)^4 \log (1-a c-b c x)}{16 e}-\frac{(b d-a e)^4 \text{Li}_2(c (a+b x))}{4 b^4 e}+\frac{(d+e x)^4 \text{Li}_2(c (a+b x))}{4 e}\\ \end{align*}
Mathematica [A] time = 0.562469, size = 485, normalized size = 0.8 \[ \frac{-144 c^4 \left (6 a^2 b^2 d^2 e-4 a^3 b d e^2+a^4 e^3-4 a b^3 d^3-b^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right ) \text{PolyLog}(2,c (a+b x))+b c \left (-6 a^2 c^2 e^2 x (b c (176 d+13 e x)+46 e)+300 a^3 c^3 e^3 x+4 a c \left (b^2 c^2 \left (324 d^2 e x-144 d^3+60 d e^2 x^2+7 e^3 x^3\right )+3 b c e^2 x (56 d+5 e x)+39 e^3 x\right )+576 b^2 c^2 d^3 (a c+b c x-1) \log (1-c (a+b x))-x \left (b^3 c^3 \left (216 d^2 e x+576 d^3+64 d e^2 x^2+9 e^3 x^3\right )+12 b^2 c^2 e \left (36 d^2+8 d e x+e^2 x^2\right )+6 b c e^2 (32 d+3 e x)+36 e^3\right )\right )+12 e (a c+b c x-1) \log (-a c-b c x+1) \left (b c e \left (8 d \left (11 a^2 c^2-7 a c+2\right )+e x \left (13 a^2 c^2-10 a c+3\right )\right )+e^2 \left (-25 a^3 c^3+23 a^2 c^2-13 a c+3\right )+b^2 c^2 \left (-36 d^2 (3 a c-1)-8 d e x (5 a c-2)+e^2 x^2 (3-7 a c)\right )+b^3 c^3 x \left (36 d^2+16 d e x+3 e^2 x^2\right )\right )}{576 b^4 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 1177, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04257, size = 919, normalized size = 1.52 \begin{align*} -\frac{{\left (4 \, a b^{3} d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a^{3} b d e^{2} - a^{4} e^{3}\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 )}}{4 \, b^{4}} - \frac{9 \, b^{4} c^{4} e^{3} x^{4} + 4 \,{\left (16 \, b^{4} c^{4} d e^{2} -{\left (7 \, a b^{3} c^{4} - 3 \, b^{3} c^{3}\right )} e^{3}\right )} x^{3} + 6 \,{\left (36 \, b^{4} c^{4} d^{2} e - 8 \,{\left (5 \, a b^{3} c^{4} - 2 \, b^{3} c^{3}\right )} d e^{2} +{\left (13 \, a^{2} b^{2} c^{4} - 10 \, a b^{2} c^{3} + 3 \, b^{2} c^{2}\right )} e^{3}\right )} x^{2} + 12 \,{\left (48 \, b^{4} c^{4} d^{3} - 36 \,{\left (3 \, a b^{3} c^{4} - b^{3} c^{3}\right )} d^{2} e + 8 \,{\left (11 \, a^{2} b^{2} c^{4} - 7 \, a b^{2} c^{3} + 2 \, b^{2} c^{2}\right )} d e^{2} -{\left (25 \, a^{3} b c^{4} - 23 \, a^{2} b c^{3} + 13 \, a b c^{2} - 3 \, b c\right )} e^{3}\right )} x - 144 \,{\left (b^{4} c^{4} e^{3} x^{4} + 4 \, b^{4} c^{4} d e^{2} x^{3} + 6 \, b^{4} c^{4} d^{2} e x^{2} + 4 \, b^{4} c^{4} d^{3} x\right )}{\rm Li}_2\left (b c x + a c\right ) - 12 \,{\left (3 \, b^{4} c^{4} e^{3} x^{4} + 48 \,{\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} - 36 \,{\left (3 \, a^{2} b^{2} c^{4} - 4 \, a b^{2} c^{3} + b^{2} c^{2}\right )} d^{2} e + 8 \,{\left (11 \, a^{3} b c^{4} - 18 \, a^{2} b c^{3} + 9 \, a b c^{2} - 2 \, b c\right )} d e^{2} -{\left (25 \, a^{4} c^{4} - 48 \, a^{3} c^{3} + 36 \, a^{2} c^{2} - 16 \, a c + 3\right )} e^{3} + 4 \,{\left (4 \, b^{4} c^{4} d e^{2} - a b^{3} c^{4} e^{3}\right )} x^{3} + 6 \,{\left (6 \, b^{4} c^{4} d^{2} e - 4 \, a b^{3} c^{4} d e^{2} + a^{2} b^{2} c^{4} e^{3}\right )} x^{2} + 12 \,{\left (4 \, b^{4} c^{4} d^{3} - 6 \, a b^{3} c^{4} d^{2} e + 4 \, a^{2} b^{2} c^{4} d e^{2} - a^{3} b c^{4} e^{3}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37849, size = 1339, normalized size = 2.21 \begin{align*} -\frac{9 \, b^{4} c^{4} e^{3} x^{4} + 4 \,{\left (16 \, b^{4} c^{4} d e^{2} -{\left (7 \, a b^{3} c^{4} - 3 \, b^{3} c^{3}\right )} e^{3}\right )} x^{3} + 6 \,{\left (36 \, b^{4} c^{4} d^{2} e - 8 \,{\left (5 \, a b^{3} c^{4} - 2 \, b^{3} c^{3}\right )} d e^{2} +{\left (13 \, a^{2} b^{2} c^{4} - 10 \, a b^{2} c^{3} + 3 \, b^{2} c^{2}\right )} e^{3}\right )} x^{2} + 12 \,{\left (48 \, b^{4} c^{4} d^{3} - 36 \,{\left (3 \, a b^{3} c^{4} - b^{3} c^{3}\right )} d^{2} e + 8 \,{\left (11 \, a^{2} b^{2} c^{4} - 7 \, a b^{2} c^{3} + 2 \, b^{2} c^{2}\right )} d e^{2} -{\left (25 \, a^{3} b c^{4} - 23 \, a^{2} b c^{3} + 13 \, a b c^{2} - 3 \, b c\right )} e^{3}\right )} x - 144 \,{\left (b^{4} c^{4} e^{3} x^{4} + 4 \, b^{4} c^{4} d e^{2} x^{3} + 6 \, b^{4} c^{4} d^{2} e x^{2} + 4 \, b^{4} c^{4} d^{3} x + 4 \, a b^{3} c^{4} d^{3} - 6 \, a^{2} b^{2} c^{4} d^{2} e + 4 \, a^{3} b c^{4} d e^{2} - a^{4} c^{4} e^{3}\right )}{\rm Li}_2\left (b c x + a c\right ) - 12 \,{\left (3 \, b^{4} c^{4} e^{3} x^{4} + 48 \,{\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} - 36 \,{\left (3 \, a^{2} b^{2} c^{4} - 4 \, a b^{2} c^{3} + b^{2} c^{2}\right )} d^{2} e + 8 \,{\left (11 \, a^{3} b c^{4} - 18 \, a^{2} b c^{3} + 9 \, a b c^{2} - 2 \, b c\right )} d e^{2} -{\left (25 \, a^{4} c^{4} - 48 \, a^{3} c^{3} + 36 \, a^{2} c^{2} - 16 \, a c + 3\right )} e^{3} + 4 \,{\left (4 \, b^{4} c^{4} d e^{2} - a b^{3} c^{4} e^{3}\right )} x^{3} + 6 \,{\left (6 \, b^{4} c^{4} d^{2} e - 4 \, a b^{3} c^{4} d e^{2} + a^{2} b^{2} c^{4} e^{3}\right )} x^{2} + 12 \,{\left (4 \, b^{4} c^{4} d^{3} - 6 \, a b^{3} c^{4} d^{2} e + 4 \, a^{2} b^{2} c^{4} d e^{2} - a^{3} b c^{4} e^{3}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \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 )}^{3}{\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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