Optimal. Leaf size=251 \[ -\frac{3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}+\frac{c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac{c^3 x^3}{3 e^4} \]
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Rubi [A] time = 0.296033, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}+\frac{c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac{c^3 x^3}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac{c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )}{e^6}-\frac{c^2 (4 c d-3 b e) x}{e^5}+\frac{c^3 x^2}{e^4}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^4}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^3}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^2}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right ) x}{e^6}-\frac{c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac{c^3 x^3}{3 e^4}-\frac{\left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^2}-\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.115325, size = 260, normalized size = 1.04 \[ \frac{-\frac{18 \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )}{d+e x}+6 c e x \left (3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right )-6 (2 c d-b e) \log (d+e x) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )-\frac{2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+\frac{9 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c^2 e^2 x^2 (3 b e-4 c d)+2 c^3 e^3 x^3}{6 e^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 653, 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 = 1.08161, size = 581, normalized size = 2.31 \begin{align*} -\frac{74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 78 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 11 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 18 \,{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} +{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \,{\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + a^{2} b e^{6} + 20 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{6 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{2 \, c^{3} e^{2} x^{3} - 3 \,{\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \,{\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x}{6 \, e^{6}} - \frac{{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04135, size = 1393, normalized size = 5.55 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} - 78 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 11 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \,{\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} +{\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \,{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 3 \,{\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 6 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 3 \,{\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 54 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 9 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 6 \,{\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} +{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \,{\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} + 3 \,{\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 34.2345, size = 498, normalized size = 1.98 \begin{align*} \frac{c^{3} x^{3}}{3 e^{4}} - \frac{2 a^{3} e^{6} + 3 a^{2} b d e^{5} + 6 a^{2} c d^{2} e^{4} + 6 a b^{2} d^{2} e^{4} - 66 a b c d^{3} e^{3} + 78 a c^{2} d^{4} e^{2} - 11 b^{3} d^{3} e^{3} + 78 b^{2} c d^{4} e^{2} - 141 b c^{2} d^{5} e + 74 c^{3} d^{6} + x^{2} \left (18 a^{2} c e^{6} + 18 a b^{2} e^{6} - 108 a b c d e^{5} + 108 a c^{2} d^{2} e^{4} - 18 b^{3} d e^{5} + 108 b^{2} c d^{2} e^{4} - 180 b c^{2} d^{3} e^{3} + 90 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} b e^{6} + 18 a^{2} c d e^{5} + 18 a b^{2} d e^{5} - 162 a b c d^{2} e^{4} + 180 a c^{2} d^{3} e^{3} - 27 b^{3} d^{2} e^{4} + 180 b^{2} c d^{3} e^{3} - 315 b c^{2} d^{4} e^{2} + 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac{x^{2} \left (3 b c^{2} e - 4 c^{3} d\right )}{2 e^{5}} + \frac{x \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 12 b c^{2} d e + 10 c^{3} d^{2}\right )}{e^{6}} + \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11308, size = 572, normalized size = 2.28 \begin{align*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c^{3} x^{3} e^{8} - 12 \, c^{3} d x^{2} e^{7} + 60 \, c^{3} d^{2} x e^{6} + 9 \, b c^{2} x^{2} e^{8} - 72 \, b c^{2} d x e^{7} + 18 \, b^{2} c x e^{8} + 18 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} + 78 \, a c^{2} d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} - 66 \, a b c d^{3} e^{3} + 6 \, a b^{2} d^{2} e^{4} + 6 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 18 \,{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} + 6 \, a c^{2} d^{2} e^{4} - b^{3} d e^{5} - 6 \, a b c d e^{5} + a b^{2} e^{6} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} + 20 \, a c^{2} d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4} - 18 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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