Optimal. Leaf size=286 \[ \frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]
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Rubi [A] time = 0.409231, antiderivative size = 284, normalized size of antiderivative = 0.99, 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} \[ \frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]
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)^4} \, dx &=\int \left (\frac{c (-4 B c d+2 b B e+A c e)}{e^5}+\frac{B c^2 x}{e^4}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^4}+\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)^3}+\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)^2}+\frac{-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{c (4 B c d-2 b B e-A c e) x}{e^5}+\frac{B c^2 x^2}{2 e^4}+\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{3 e^6 (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 )}{2 e^6 (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^6 (d+e x)}-\frac{\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.144668, size = 263, normalized size = 0.92 \[ \frac{-\frac{6 \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )}{d+e x}+6 \log (d+e x) \left (B \left (2 c e (a e-4 b d)+b^2 e^2+10 c^2 d^2\right )+2 A c e (b e-2 c d)\right )+\frac{2 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-\frac{3 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{(d+e x)^2}+6 c e x (A c e+2 b B e-4 B c d)+3 B c^2 e^2 x^2}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 690, normalized size = 2.4 \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 = 1.04748, size = 552, normalized size = 1.93 \begin{align*} \frac{47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\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} + 6 \,{\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} + 3 \,{\left (35 \, B c^{2} d^{4} e - 20 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\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 (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B c^{2} e x^{2} - 2 \,{\left (4 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )} x}{2 \, e^{5}} + \frac{{\left (10 \, B c^{2} d^{2} - 4 \,{\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 )} \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 = 1.52217, size = 1324, normalized size = 4.63 \begin{align*} \frac{3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\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} - 3 \,{\left (5 \, B c^{2} d e^{4} - 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B c^{2} d^{2} e^{3} - 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B c^{2} d^{3} e^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 6 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, B c^{2} d^{4} e - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\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} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} +{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \,{\left (10 \, B c^{2} d^{3} e^{2} - 4 \,{\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^{4}\right )} x^{2} + 3 \,{\left (10 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \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 [A] time = 1.17962, size = 572, normalized size = 2. \begin{align*}{\left (10 \, B c^{2} d^{2} - 8 \, B b c d e - 4 \, A c^{2} d e + B b^{2} e^{2} + 2 \, B a c e^{2} + 2 \, A b c e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B c^{2} d^{5} - 52 \, B b c d^{4} e - 26 \, A c^{2} d^{4} e + 11 \, B b^{2} d^{3} e^{2} + 22 \, B a c d^{3} e^{2} + 22 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a b d e^{4} - 2 \, A a^{2} e^{5} + 6 \,{\left (10 \, B c^{2} d^{3} e^{2} - 12 \, B b c d^{2} e^{3} - 6 \, A c^{2} d^{2} e^{3} + 3 \, B b^{2} d e^{4} + 6 \, B a c d e^{4} + 6 \, A b c d e^{4} - 2 \, B a b e^{5} - A b^{2} e^{5} - 2 \, A a c e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 40 \, B b c d^{3} e^{2} - 20 \, A c^{2} d^{3} e^{2} + 9 \, B b^{2} d^{2} e^{3} + 18 \, B a c d^{2} e^{3} + 18 \, 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 )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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