Optimal. Leaf size=605 \[ -\frac {(-a c e+b c d+e)^4 \log (-a c-b c x+1)}{16 b^4 c^4 e}-\frac {(b d-a e) (-a c e+b c d+e)^3 \log (-a c-b c x+1)}{12 b^4 c^3 e}-\frac {(b d-a e)^2 (-a c e+b c d+e)^2 \log (-a c-b c x+1)}{8 b^4 c^2 e}-\frac {(b d-a e)^4 \text {Li}_2(c (a+b x))}{4 b^4 e}-\frac {(-a c-b c x+1) (b d-a e)^3 \log (-a c-b c x+1)}{4 b^4 c}-\frac {x (-a c e+b c d+e)^3}{16 b^3 c^3}-\frac {x (b d-a e) (-a c e+b c d+e)^2}{12 b^3 c^2}-\frac {x (b d-a e)^2 (-a c e+b c d+e)}{8 b^3 c}-\frac {x (b d-a e)^3}{4 b^3}-\frac {(d+e x)^2 (-a c e+b c d+e)^2}{32 b^2 c^2 e}-\frac {(d+e x)^2 (b d-a e) (-a c e+b c d+e)}{24 b^2 c e}+\frac {(d+e x)^2 (b d-a e)^2 \log (-a c-b c x+1)}{8 b^2 e}-\frac {(d+e x)^2 (b d-a e)^2}{16 b^2 e}+\frac {(d+e x)^4 \text {Li}_2(c (a+b x))}{4 e}-\frac {(d+e x)^3 (-a c e+b c d+e)}{48 b c e}+\frac {(d+e x)^3 (b d-a e) \log (-a c-b c x+1)}{12 b e}+\frac {(d+e x)^4 \log (-a c-b c x+1)}{16 e}-\frac {(d+e x)^3 (b d-a e)}{36 b e}-\frac {(d+e x)^4}{64 e} \]
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Rubi [A] time = 0.59, antiderivative size = 605, normalized size of antiderivative = 1.00, 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 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2395
Rule 2418
Rule 6598
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*}
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Mathematica [A] time = 0.53, size = 485, normalized size = 0.80 \[ \frac {12 e (a c+b c x-1) \log (-a c-b c x+1) \left (b c e \left (8 d \left (11 a^2 c^2-7 a c+2\right )+e x \left (13 a^2 c^2-10 a c+3\right )\right )+e^2 \left (-25 a^3 c^3+23 a^2 c^2-13 a c+3\right )+b^2 c^2 \left (-36 d^2 (3 a c-1)-8 d e x (5 a c-2)+e^2 x^2 (3-7 a c)\right )+b^3 c^3 x \left (36 d^2+16 d e x+3 e^2 x^2\right )\right )+b c \left (300 a^3 c^3 e^3 x-6 a^2 c^2 e^2 x (b c (176 d+13 e x)+46 e)+576 b^2 c^2 d^3 (a c+b c x-1) \log (1-c (a+b x))+4 a c \left (b^2 c^2 \left (-144 d^3+324 d^2 e x+60 d e^2 x^2+7 e^3 x^3\right )+3 b c e^2 x (56 d+5 e x)+39 e^3 x\right )-x \left (b^3 c^3 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )+12 b^2 c^2 e \left (36 d^2+8 d e x+e^2 x^2\right )+6 b c e^2 (32 d+3 e x)+36 e^3\right )\right )-144 c^4 \text {Li}_2(c (a+b x)) \left (a^4 e^3-4 a^3 b d e^2+6 a^2 b^2 d^2 e-4 a b^3 d^3-b^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )}{576 b^4 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 651, normalized size = 1.08 \[ -\frac {9 \, b^{4} c^{4} e^{3} x^{4} + 4 \, {\left (16 \, b^{4} c^{4} d e^{2} - {\left (7 \, a b^{3} c^{4} - 3 \, b^{3} c^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (36 \, b^{4} c^{4} d^{2} e - 8 \, {\left (5 \, a b^{3} c^{4} - 2 \, b^{3} c^{3}\right )} d e^{2} + {\left (13 \, a^{2} b^{2} c^{4} - 10 \, a b^{2} c^{3} + 3 \, b^{2} c^{2}\right )} e^{3}\right )} x^{2} + 12 \, {\left (48 \, b^{4} c^{4} d^{3} - 36 \, {\left (3 \, a b^{3} c^{4} - b^{3} c^{3}\right )} d^{2} e + 8 \, {\left (11 \, a^{2} b^{2} c^{4} - 7 \, a b^{2} c^{3} + 2 \, b^{2} c^{2}\right )} d e^{2} - {\left (25 \, a^{3} b c^{4} - 23 \, a^{2} b c^{3} + 13 \, a b c^{2} - 3 \, b c\right )} e^{3}\right )} x - 144 \, {\left (b^{4} c^{4} e^{3} x^{4} + 4 \, b^{4} c^{4} d e^{2} x^{3} + 6 \, b^{4} c^{4} d^{2} e x^{2} + 4 \, b^{4} c^{4} d^{3} x + 4 \, a b^{3} c^{4} d^{3} - 6 \, a^{2} b^{2} c^{4} d^{2} e + 4 \, a^{3} b c^{4} d e^{2} - a^{4} c^{4} e^{3}\right )} {\rm Li}_2\left (b c x + a c\right ) - 12 \, {\left (3 \, b^{4} c^{4} e^{3} x^{4} + 48 \, {\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} - 36 \, {\left (3 \, a^{2} b^{2} c^{4} - 4 \, a b^{2} c^{3} + b^{2} c^{2}\right )} d^{2} e + 8 \, {\left (11 \, a^{3} b c^{4} - 18 \, a^{2} b c^{3} + 9 \, a b c^{2} - 2 \, b c\right )} d e^{2} - {\left (25 \, a^{4} c^{4} - 48 \, a^{3} c^{3} + 36 \, a^{2} c^{2} - 16 \, a c + 3\right )} e^{3} + 4 \, {\left (4 \, b^{4} c^{4} d e^{2} - a b^{3} c^{4} e^{3}\right )} x^{3} + 6 \, {\left (6 \, b^{4} c^{4} d^{2} e - 4 \, a b^{3} c^{4} d e^{2} + a^{2} b^{2} c^{4} e^{3}\right )} x^{2} + 12 \, {\left (4 \, b^{4} c^{4} d^{3} - 6 \, a b^{3} c^{4} d^{2} e + 4 \, a^{2} b^{2} c^{4} d e^{2} - a^{3} b c^{4} e^{3}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \]
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 \[ \int {\left (e x + d\right )}^{3} {\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1177, normalized size = 1.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 681, normalized size = 1.13 \[ -\frac {{\left (4 \, a b^{3} d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a^{3} b d e^{2} - a^{4} e^{3}\right )} {\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )}}{4 \, b^{4}} - \frac {9 \, b^{4} c^{4} e^{3} x^{4} + 4 \, {\left (16 \, b^{4} c^{4} d e^{2} - {\left (7 \, a b^{3} c^{4} - 3 \, b^{3} c^{3}\right )} e^{3}\right )} x^{3} + 6 \, {\left (36 \, b^{4} c^{4} d^{2} e - 8 \, {\left (5 \, a b^{3} c^{4} - 2 \, b^{3} c^{3}\right )} d e^{2} + {\left (13 \, a^{2} b^{2} c^{4} - 10 \, a b^{2} c^{3} + 3 \, b^{2} c^{2}\right )} e^{3}\right )} x^{2} + 12 \, {\left (48 \, b^{4} c^{4} d^{3} - 36 \, {\left (3 \, a b^{3} c^{4} - b^{3} c^{3}\right )} d^{2} e + 8 \, {\left (11 \, a^{2} b^{2} c^{4} - 7 \, a b^{2} c^{3} + 2 \, b^{2} c^{2}\right )} d e^{2} - {\left (25 \, a^{3} b c^{4} - 23 \, a^{2} b c^{3} + 13 \, a b c^{2} - 3 \, b c\right )} e^{3}\right )} x - 144 \, {\left (b^{4} c^{4} e^{3} x^{4} + 4 \, b^{4} c^{4} d e^{2} x^{3} + 6 \, b^{4} c^{4} d^{2} e x^{2} + 4 \, b^{4} c^{4} d^{3} x\right )} {\rm Li}_2\left (b c x + a c\right ) - 12 \, {\left (3 \, b^{4} c^{4} e^{3} x^{4} + 48 \, {\left (a b^{3} c^{4} - b^{3} c^{3}\right )} d^{3} - 36 \, {\left (3 \, a^{2} b^{2} c^{4} - 4 \, a b^{2} c^{3} + b^{2} c^{2}\right )} d^{2} e + 8 \, {\left (11 \, a^{3} b c^{4} - 18 \, a^{2} b c^{3} + 9 \, a b c^{2} - 2 \, b c\right )} d e^{2} - {\left (25 \, a^{4} c^{4} - 48 \, a^{3} c^{3} + 36 \, a^{2} c^{2} - 16 \, a c + 3\right )} e^{3} + 4 \, {\left (4 \, b^{4} c^{4} d e^{2} - a b^{3} c^{4} e^{3}\right )} x^{3} + 6 \, {\left (6 \, b^{4} c^{4} d^{2} e - 4 \, a b^{3} c^{4} d e^{2} + a^{2} b^{2} c^{4} e^{3}\right )} x^{2} + 12 \, {\left (4 \, b^{4} c^{4} d^{3} - 6 \, a b^{3} c^{4} d^{2} e + 4 \, a^{2} b^{2} c^{4} d e^{2} - a^{3} b c^{4} e^{3}\right )} x\right )} \log \left (-b c x - a c + 1\right )}{576 \, b^{4} c^{4}} \]
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 \[ \int \mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.56, size = 1028, normalized size = 1.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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