Optimal. Leaf size=267 \[ -\frac{x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}-\frac{x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6}-\frac{c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac{B c^2 x^4}{4 e^2} \]
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Rubi [A] time = 0.490439, antiderivative size = 264, 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} \[ -\frac{x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}-\frac{x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6}-\frac{c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac{B c^2 x^4}{4 e^2} \]
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)^2} \, dx &=\int \left (\frac{-2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )+A e \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )}{e^5}+\frac{\left (-2 A c e (c d-b e)+B \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )\right ) x}{e^4}+\frac{c (-2 B c d+2 b B e+A c e) x^2}{e^3}+\frac{B c^2 x^3}{e^2}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (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^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )\right ) x}{e^5}-\frac{\left (2 A c e (c d-b e)-B \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )\right ) x^2}{2 e^4}-\frac{c (2 B c d-2 b B e-A c e) x^3}{3 e^3}+\frac{B c^2 x^4}{4 e^2}+\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)}+\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 ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.135153, size = 250, normalized size = 0.94 \[ \frac{6 e^2 x^2 \left (B \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )+2 A c e (b e-c d)\right )+12 e x \left (A e \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )-2 B (2 c d-b e) \left (e (a e-b d)+c d^2\right )\right )+\frac{12 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+12 \log (d+e x) \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )+4 c e^3 x^3 (A c e+2 b B e-2 B c d)+3 B c^2 e^4 x^4}{12 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 609, normalized size = 2.3 \begin{align*}{\frac{Ab{x}^{2}c}{{e}^{2}}}-{\frac{A{c}^{2}{x}^{2}d}{{e}^{3}}}+{\frac{3\,B{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{4}}}+3\,{\frac{A{c}^{2}{d}^{2}x}{{e}^{4}}}+{\frac{2\,B{x}^{3}bc}{3\,{e}^{2}}}-{\frac{2\,B{c}^{2}{x}^{3}d}{3\,{e}^{3}}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ) B{b}^{2}}{{e}^{4}}}+5\,{\frac{{d}^{4}\ln \left ( ex+d \right ) B{c}^{2}}{{e}^{6}}}-2\,{\frac{B{b}^{2}dx}{{e}^{3}}}-4\,{\frac{B{c}^{2}{d}^{3}x}{{e}^{5}}}+{\frac{B{d}^{3}{b}^{2}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{B{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-2\,{\frac{d\ln \left ( ex+d \right ) A{b}^{2}}{{e}^{3}}}-4\,{\frac{{d}^{3}\ln \left ( ex+d \right ) A{c}^{2}}{{e}^{5}}}-{\frac{{d}^{2}A{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{A{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+2\,{\frac{\ln \left ( ex+d \right ) Aab}{{e}^{2}}}+2\,{\frac{aAcx}{{e}^{2}}}+{\frac{aBc{x}^{2}}{{e}^{2}}}+2\,{\frac{abBx}{{e}^{2}}}+{\frac{{b}^{2}B{x}^{2}}{2\,{e}^{2}}}+{\frac{B{c}^{2}{x}^{4}}{4\,{e}^{2}}}-4\,{\frac{aBdcx}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ) Aacd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) Bac{d}^{2}}{{e}^{4}}}-2\,{\frac{{d}^{2}Aac}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{aBc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{B{x}^{2}bcd}{{e}^{3}}}-4\,{\frac{Abcdx}{{e}^{3}}}+6\,{\frac{Bbc{d}^{2}x}{{e}^{4}}}+6\,{\frac{{d}^{2}\ln \left ( ex+d \right ) Abc}{{e}^{4}}}-8\,{\frac{{d}^{3}\ln \left ( ex+d \right ) Bbc}{{e}^{5}}}+2\,{\frac{A{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{{d}^{4}Bbc}{{e}^{5} \left ( ex+d \right ) }}-4\,{\frac{\ln \left ( ex+d \right ) Babd}{{e}^{3}}}+2\,{\frac{Adab}{{e}^{2} \left ( ex+d \right ) }}-2\,{\frac{abB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{A{a}^{2}}{e \left ( ex+d \right ) }}+{\frac{\ln \left ( ex+d \right ) B{a}^{2}}{{e}^{2}}}+{\frac{A{b}^{2}x}{{e}^{2}}}+{\frac{A{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{Bd{a}^{2}}{{e}^{2} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996193, size = 522, normalized size = 1.96 \begin{align*} \frac{B c^{2} d^{5} - A a^{2} e^{5} -{\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 (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}}{e^{7} x + d e^{6}} + \frac{3 \, B c^{2} e^{3} x^{4} - 4 \,{\left (2 \, B c^{2} d e^{2} -{\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B c^{2} d^{2} e - 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{3}\right )} x^{2} - 12 \,{\left (4 \, B c^{2} d^{3} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 2 \,{\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 )} x}{12 \, e^{5}} + \frac{{\left (5 \, B c^{2} d^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\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 = 1.28211, size = 1233, normalized size = 4.62 \begin{align*} \frac{3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 12 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 12 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} -{\left (5 \, B c^{2} d e^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (5 \, B c^{2} d^{2} e^{3} - 4 \,{\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} - 6 \,{\left (5 \, B c^{2} d^{3} e^{2} - 4 \,{\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} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 12 \,{\left (4 \, B c^{2} d^{4} e - 3 \,{\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} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x + 12 \,{\left (5 \, B c^{2} d^{5} - 4 \,{\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} - 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} +{\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\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.69562, size = 432, normalized size = 1.62 \begin{align*} \frac{B c^{2} x^{4}}{4 e^{2}} + \frac{- A a^{2} e^{5} + 2 A a b d e^{4} - 2 A a c d^{2} e^{3} - A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B a^{2} d e^{4} - 2 B a b d^{2} e^{3} + 2 B a c d^{3} e^{2} + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (A c^{2} e + 2 B b c e - 2 B c^{2} d\right )}{3 e^{3}} + \frac{x^{2} \left (2 A b c e^{2} - 2 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 4 B b c d e + 3 B c^{2} d^{2}\right )}{2 e^{4}} + \frac{x \left (2 A a c e^{3} + A b^{2} e^{3} - 4 A b c d e^{2} + 3 A c^{2} d^{2} e + 2 B a b e^{3} - 4 B a c d e^{2} - 2 B b^{2} d e^{2} + 6 B b c d^{2} e - 4 B c^{2} d^{3}\right )}{e^{5}} + \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (2 A b e^{2} - 4 A c d e + B a e^{2} - 3 B b d e + 5 B c d^{2}\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 [B] time = 1.21749, size = 744, normalized size = 2.79 \begin{align*} \frac{1}{12} \,{\left (3 \, B c^{2} - \frac{4 \,{\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{6 \,{\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, B a c e^{4} + 2 \, A b c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{12 \,{\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, B a c d e^{5} + 6 \, A b c d e^{5} - 2 \, B a b e^{6} - A b^{2} e^{6} - 2 \, A a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{4} e^{\left (-6\right )} -{\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, B a c d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} + B a^{2} e^{4} + 2 \, A a b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c^{2} d^{5} e^{4}}{x e + d} - \frac{2 \, B b c d^{4} e^{5}}{x e + d} - \frac{A c^{2} d^{4} e^{5}}{x e + d} + \frac{B b^{2} d^{3} e^{6}}{x e + d} + \frac{2 \, B a c d^{3} e^{6}}{x e + d} + \frac{2 \, A b c d^{3} e^{6}}{x e + d} - \frac{2 \, B a b d^{2} e^{7}}{x e + d} - \frac{A b^{2} d^{2} e^{7}}{x e + d} - \frac{2 \, A a c d^{2} e^{7}}{x e + d} + \frac{B a^{2} d e^{8}}{x e + d} + \frac{2 \, A a b d e^{8}}{x e + d} - \frac{A a^{2} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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