Optimal. Leaf size=281 \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac{\log (d+e x) \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.439505, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac{\log (d+e x) \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{-A c e (3 c d-2 b e)+B \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5}+\frac{c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac{B c^2 x^2}{e^3}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^3}+\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^2}+\frac{-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) x}{e^5}-\frac{c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac{B c^2 x^3}{3 e^3}+\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{2 e^6 (d+e x)^2}-\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^6 (d+e x)}-\frac{\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.151491, size = 262, normalized size = 0.93 \[ \frac{6 e x \left (B \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+A c e (2 b e-3 c d)\right )+6 \log (d+e x) \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )-\frac{6 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{d+e x}+\frac{3 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)+2 B c^2 e^3 x^3}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 654, normalized size = 2.3 \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 = 1.01573, size = 536, normalized size = 1.91 \begin{align*} -\frac{9 \, B c^{2} d^{5} + A a^{2} e^{5} - 7 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 3 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 2 \,{\left (5 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B c^{2} e^{2} x^{3} - 3 \,{\left (3 \, B c^{2} d e -{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B c^{2} d^{2} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{{\left (10 \, B c^{2} d^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3437, size = 1332, normalized size = 4.74 \begin{align*} \frac{2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 21 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} + 9 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} -{\left (5 \, B c^{2} d e^{4} - 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (10 \, B c^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B c^{2} d^{3} e^{2} - 11 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B c^{2} d^{4} e +{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} -{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 6 \,{\left (10 \, B c^{2} d^{5} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (10 \, B c^{2} d^{3} e^{2} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 2 \,{\left (10 \, B c^{2} d^{4} e - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.9181, size = 527, normalized size = 1.88 \begin{align*} \frac{B c^{2} x^{3}}{3 e^{3}} - \frac{A a^{2} e^{5} + 2 A a b d e^{4} - 6 A a c d^{2} e^{3} - 3 A b^{2} d^{2} e^{3} + 10 A b c d^{3} e^{2} - 7 A c^{2} d^{4} e + B a^{2} d e^{4} - 6 B a b d^{2} e^{3} + 10 B a c d^{3} e^{2} + 5 B b^{2} d^{3} e^{2} - 14 B b c d^{4} e + 9 B c^{2} d^{5} + x \left (4 A a b e^{5} - 8 A a c d e^{4} - 4 A b^{2} d e^{4} + 12 A b c d^{2} e^{3} - 8 A c^{2} d^{3} e^{2} + 2 B a^{2} e^{5} - 8 B a b d e^{4} + 12 B a c d^{2} e^{3} + 6 B b^{2} d^{2} e^{3} - 16 B b c d^{3} e^{2} + 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A c^{2} e + 2 B b c e - 3 B c^{2} d\right )}{2 e^{4}} + \frac{x \left (2 A b c e^{2} - 3 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 6 B b c d e + 6 B c^{2} d^{2}\right )}{e^{5}} + \frac{\left (2 A a c e^{3} + A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e + 2 B a b e^{3} - 6 B a c d e^{2} - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right ) \log{\left (d + e x \right )}}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21908, size = 581, normalized size = 2.07 \begin{align*} -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, B a c d e^{2} + 6 \, A b c d e^{2} - 2 \, B a b e^{3} - A b^{2} e^{3} - 2 \, A a c e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, B a c x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, B a c d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 2 \, A a b d e^{4} + A a^{2} e^{5} + 2 \,{\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, B a c d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} + B a^{2} e^{5} + 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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