Optimal. Leaf size=396 \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac{c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}-\frac{c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac{c^4 x^4}{2 e^4} \]
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Rubi [A] time = 0.512417, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac{c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}-\frac{c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac{c^4 x^4}{2 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac{c \left (-40 c^3 d^3+5 b^3 e^3+2 c^2 d e (35 b d-12 a e)-3 b c e^2 (12 b d-5 a e)\right )}{e^7}+\frac{c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x}{e^6}-\frac{c^3 (8 c d-7 b e) x^2}{e^5}+\frac{2 c^4 x^3}{e^4}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^4}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^7 (d+e x)^2}+\frac{70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (40 c^3 d^3-5 b^3 e^3-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)\right ) x}{e^7}+\frac{c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2}{2 e^6}-\frac{c^3 (8 c d-7 b e) x^3}{3 e^5}+\frac{c^4 x^4}{2 e^4}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^8 (d+e x)^2}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^8 (d+e x)}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^8}\\ \end{align*}
Mathematica [A] time = 0.170926, size = 404, normalized size = 1.02 \[ \frac{\frac{18 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{d+e x}+6 \log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )+3 c^2 e^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )-6 c e x \left (2 c^2 d e (12 a e-35 b d)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )-\frac{3 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+\frac{2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}-2 c^3 e^3 x^3 (8 c d-7 b e)+3 c^4 e^4 x^4}{6 e^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 1023, normalized size = 2.6 \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 = 1.18169, size = 905, normalized size = 2.29 \begin{align*} \frac{214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e - 2 \, a^{3} b e^{7} + 141 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 130 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 11 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 6 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} -{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} + 18 \,{\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 10 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 10 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} -{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 3 \,{\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 105 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 100 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 9 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 6 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} -{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x}{6 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac{3 \, c^{4} e^{3} x^{4} - 2 \,{\left (8 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} x^{3} + 3 \,{\left (20 \, c^{4} d^{2} e - 28 \, b c^{3} d e^{2} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{2} - 6 \,{\left (40 \, c^{4} d^{3} - 70 \, b c^{3} d^{2} e + 12 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x}{6 \, e^{7}} + \frac{{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54624, size = 2175, normalized size = 5.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 91.6159, size = 814, normalized size = 2.06 \begin{align*} \frac{c^{4} x^{4}}{2 e^{4}} - \frac{2 a^{3} b e^{7} + 2 a^{3} c d e^{6} + 3 a^{2} b^{2} d e^{6} + 18 a^{2} b c d^{2} e^{5} - 66 a^{2} c^{2} d^{3} e^{4} + 6 a b^{3} d^{2} e^{5} - 132 a b^{2} c d^{3} e^{4} + 390 a b c^{2} d^{4} e^{3} - 282 a c^{3} d^{5} e^{2} - 11 b^{4} d^{3} e^{4} + 130 b^{3} c d^{4} e^{3} - 423 b^{2} c^{2} d^{5} e^{2} + 518 b c^{3} d^{6} e - 214 c^{4} d^{7} + x^{2} \left (54 a^{2} b c e^{7} - 108 a^{2} c^{2} d e^{6} + 18 a b^{3} e^{7} - 216 a b^{2} c d e^{6} + 540 a b c^{2} d^{2} e^{5} - 360 a c^{3} d^{3} e^{4} - 18 b^{4} d e^{6} + 180 b^{3} c d^{2} e^{5} - 540 b^{2} c^{2} d^{3} e^{4} + 630 b c^{3} d^{4} e^{3} - 252 c^{4} d^{5} e^{2}\right ) + x \left (6 a^{3} c e^{7} + 9 a^{2} b^{2} e^{7} + 54 a^{2} b c d e^{6} - 162 a^{2} c^{2} d^{2} e^{5} + 18 a b^{3} d e^{6} - 324 a b^{2} c d^{2} e^{5} + 900 a b c^{2} d^{3} e^{4} - 630 a c^{3} d^{4} e^{3} - 27 b^{4} d^{2} e^{5} + 300 b^{3} c d^{3} e^{4} - 945 b^{2} c^{2} d^{4} e^{3} + 1134 b c^{3} d^{5} e^{2} - 462 c^{4} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} + \frac{x^{3} \left (7 b c^{3} e - 8 c^{4} d\right )}{3 e^{5}} + \frac{x^{2} \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 28 b c^{3} d e + 20 c^{4} d^{2}\right )}{2 e^{6}} + \frac{x \left (15 a b c^{2} e^{3} - 24 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 36 b^{2} c^{2} d e^{2} + 70 b c^{3} d^{2} e - 40 c^{4} d^{3}\right )}{e^{7}} + \frac{\left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right ) \log{\left (d + e x \right )}}{e^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18968, size = 915, normalized size = 2.31 \begin{align*}{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (3 \, c^{4} x^{4} e^{12} - 16 \, c^{4} d x^{3} e^{11} + 60 \, c^{4} d^{2} x^{2} e^{10} - 240 \, c^{4} d^{3} x e^{9} + 14 \, b c^{3} x^{3} e^{12} - 84 \, b c^{3} d x^{2} e^{11} + 420 \, b c^{3} d^{2} x e^{10} + 27 \, b^{2} c^{2} x^{2} e^{12} + 18 \, a c^{3} x^{2} e^{12} - 216 \, b^{2} c^{2} d x e^{11} - 144 \, a c^{3} d x e^{11} + 30 \, b^{3} c x e^{12} + 90 \, a b c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac{{\left (214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e + 423 \, b^{2} c^{2} d^{5} e^{2} + 282 \, a c^{3} d^{5} e^{2} - 130 \, b^{3} c d^{4} e^{3} - 390 \, a b c^{2} d^{4} e^{3} + 11 \, b^{4} d^{3} e^{4} + 132 \, a b^{2} c d^{3} e^{4} + 66 \, a^{2} c^{2} d^{3} e^{4} - 6 \, a b^{3} d^{2} e^{5} - 18 \, a^{2} b c d^{2} e^{5} - 3 \, a^{2} b^{2} d e^{6} - 2 \, a^{3} c d e^{6} - 2 \, a^{3} b e^{7} + 18 \,{\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 30 \, b^{2} c^{2} d^{3} e^{4} + 20 \, a c^{3} d^{3} e^{4} - 10 \, b^{3} c d^{2} e^{5} - 30 \, a b c^{2} d^{2} e^{5} + b^{4} d e^{6} + 12 \, a b^{2} c d e^{6} + 6 \, a^{2} c^{2} d e^{6} - a b^{3} e^{7} - 3 \, a^{2} b c e^{7}\right )} x^{2} + 3 \,{\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 315 \, b^{2} c^{2} d^{4} e^{3} + 210 \, a c^{3} d^{4} e^{3} - 100 \, b^{3} c d^{3} e^{4} - 300 \, a b c^{2} d^{3} e^{4} + 9 \, b^{4} d^{2} e^{5} + 108 \, a b^{2} c d^{2} e^{5} + 54 \, a^{2} c^{2} d^{2} e^{5} - 6 \, a b^{3} d e^{6} - 18 \, a^{2} b c d e^{6} - 3 \, a^{2} b^{2} e^{7} - 2 \, a^{3} c e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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