Optimal. Leaf size=257 \[ -\frac{x^3 \left (A c e (c d-2 b e)-B \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{3 e^3}-\frac{x^2 \left (B (c d-b e) \left (c d^2-e (b d-2 a e)\right )-A e \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{2 e^4}-\frac{x \left (A e (c d-b e) \left (c d^2-e (b d-2 a e)\right )-B \left (c d^2-e (b d-a e)\right )^2\right )}{e^5}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{c x^4 (-A c e-2 b B e+B c d)}{4 e^2}+\frac{B c^2 x^5}{5 e} \]
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Rubi [A] time = 0.537828, antiderivative size = 257, 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} \[ -\frac{x^3 \left (A c e (c d-2 b e)-B \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{3 e^3}-\frac{x^2 \left (B (c d-b e) \left (c d^2-e (b d-2 a e)\right )-A e \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{2 e^4}-\frac{x \left (A e (c d-b e) \left (c d^2-e (b d-2 a e)\right )-B \left (c d^2-e (b d-a e)\right )^2\right )}{e^5}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{c x^4 (-A c e-2 b B e+B c d)}{4 e^2}+\frac{B c^2 x^5}{5 e} \]
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} \, dx &=\int \left (\frac{-A e (c d-b e) \left (c d^2-e (b d-2 a e)\right )+B \left (c d^2-e (b d-a e)\right )^2}{e^5}+\frac{\left (-B (c d-b e) \left (c d^2-e (b d-2 a e)\right )+A e \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x}{e^4}+\frac{\left (-A c e (c d-2 b e)+B \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x^2}{e^3}+\frac{c (-B c d+2 b B e+A c e) x^3}{e^2}+\frac{B c^2 x^4}{e}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (A e (c d-b e) \left (c d^2-e (b d-2 a e)\right )-B \left (c d^2-e (b d-a e)\right )^2\right ) x}{e^5}-\frac{\left (B (c d-b e) \left (c d^2-e (b d-2 a e)\right )-A e \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x^2}{2 e^4}-\frac{\left (A c e (c d-2 b e)-B \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x^3}{3 e^3}-\frac{c (B c d-2 b B e-A c e) x^4}{4 e^2}+\frac{B c^2 x^5}{5 e}-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.207958, size = 298, normalized size = 1.16 \[ \frac{e x \left (B \left (10 e^2 \left (6 a^2 e^2+6 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+10 c e \left (2 a e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+c^2 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+5 A e \left (4 c e \left (3 a e (e x-2 d)+b \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+6 b e^2 (4 a e-2 b d+b e x)+c^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )-60 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2}{60 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 558, normalized size = 2.2 \begin{align*}{\frac{B{c}^{2}{x}^{5}}{5\,e}}-2\,{\frac{abBdx}{{e}^{2}}}+2\,{\frac{aBc{d}^{2}x}{{e}^{3}}}+2\,{\frac{A{d}^{2}bcx}{{e}^{3}}}+{\frac{B{x}^{2}bc{d}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ) aBc{d}^{3}}{{e}^{4}}}+2\,{\frac{\ln \left ( ex+d \right ) Bbc{d}^{4}}{{e}^{5}}}+2\,{\frac{\ln \left ( ex+d \right ) A{d}^{2}ac}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ) Abc{d}^{3}}{{e}^{4}}}+2\,{\frac{\ln \left ( ex+d \right ) aBb{d}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ) Aabd}{{e}^{2}}}-{\frac{2\,B{x}^{3}bcd}{3\,{e}^{2}}}-{\frac{aBc{x}^{2}d}{{e}^{2}}}-{\frac{Ab{x}^{2}cd}{{e}^{2}}}-2\,{\frac{Bcb{d}^{3}x}{{e}^{4}}}-2\,{\frac{aAcdx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) A{a}^{2}}{e}}+{\frac{A{b}^{2}{x}^{2}}{2\,e}}+{\frac{{a}^{2}Bx}{e}}+{\frac{A{c}^{2}{x}^{4}}{4\,e}}+{\frac{B{x}^{3}{b}^{2}}{3\,e}}-{\frac{B{c}^{2}{x}^{4}d}{4\,{e}^{2}}}+{\frac{2\,Ab{x}^{3}c}{3\,e}}+{\frac{aAc{x}^{2}}{e}}-{\frac{B{c}^{2}{x}^{2}{d}^{3}}{2\,{e}^{4}}}+{\frac{B{c}^{2}{d}^{4}x}{{e}^{5}}}+{\frac{B{b}^{2}{d}^{2}x}{{e}^{3}}}-{\frac{A{b}^{2}dx}{{e}^{2}}}+{\frac{B{x}^{4}bc}{2\,e}}-{\frac{A{c}^{2}{x}^{3}d}{3\,{e}^{2}}}+{\frac{2\,aBc{x}^{3}}{3\,e}}+{\frac{B{c}^{2}{x}^{3}{d}^{2}}{3\,{e}^{3}}}-{\frac{{b}^{2}B{x}^{2}d}{2\,{e}^{2}}}+2\,{\frac{aAbx}{e}}-{\frac{A{d}^{3}{c}^{2}x}{{e}^{4}}}+{\frac{B{x}^{2}ab}{e}}+{\frac{\ln \left ( ex+d \right ) A{d}^{4}{c}^{2}}{{e}^{5}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) A{b}^{2}{d}^{2}}{{e}^{3}}}+{\frac{A{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) B{b}^{2}{d}^{3}}{{e}^{4}}}-{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{5}}{{e}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982496, size = 510, normalized size = 1.98 \begin{align*} \frac{12 \, B c^{2} e^{4} x^{5} - 15 \,{\left (B c^{2} d e^{3} -{\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{4} + 20 \,{\left (B c^{2} d^{2} e^{2} -{\left (2 \, B b c + A c^{2}\right )} d e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{4}\right )} x^{3} - 30 \,{\left (B c^{2} d^{3} e -{\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^{3} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{2} + 60 \,{\left (B c^{2} d^{4} -{\left (2 \, B b c + A c^{2}\right )} d^{3} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{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 )} x}{60 \, e^{5}} - \frac{{\left (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}\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 [A] time = 1.19892, size = 801, normalized size = 3.12 \begin{align*} \frac{12 \, B c^{2} e^{5} x^{5} - 15 \,{\left (B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (B c^{2} d^{2} e^{3} -{\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} - 30 \,{\left (B c^{2} d^{3} e^{2} -{\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} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 60 \,{\left (B c^{2} d^{4} e -{\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} -{\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 - 60 \,{\left (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}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.63445, size = 369, normalized size = 1.44 \begin{align*} \frac{B c^{2} x^{5}}{5 e} + \frac{x^{4} \left (A c^{2} e + 2 B b c e - B c^{2} d\right )}{4 e^{2}} + \frac{x^{3} \left (2 A b c e^{2} - A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 2 B b c d e + B c^{2} d^{2}\right )}{3 e^{3}} + \frac{x^{2} \left (2 A a c e^{3} + A b^{2} e^{3} - 2 A b c d e^{2} + A c^{2} d^{2} e + 2 B a b e^{3} - 2 B a c d e^{2} - B b^{2} d e^{2} + 2 B b c d^{2} e - B c^{2} d^{3}\right )}{2 e^{4}} + \frac{x \left (2 A a b e^{4} - 2 A a c d e^{3} - A b^{2} d e^{3} + 2 A b c d^{2} e^{2} - A c^{2} d^{3} e + B a^{2} e^{4} - 2 B a b d e^{3} + 2 B a c d^{2} e^{2} + B b^{2} d^{2} e^{2} - 2 B b c d^{3} e + B c^{2} d^{4}\right )}{e^{5}} - \frac{\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \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 = 1.15785, size = 625, normalized size = 2.43 \begin{align*} -{\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - 2 \, B a b d^{2} e^{3} - A b^{2} d^{2} e^{3} - 2 \, 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}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (12 \, B c^{2} x^{5} e^{4} - 15 \, B c^{2} d x^{4} e^{3} + 20 \, B c^{2} d^{2} x^{3} e^{2} - 30 \, B c^{2} d^{3} x^{2} e + 60 \, B c^{2} d^{4} x + 30 \, B b c x^{4} e^{4} + 15 \, A c^{2} x^{4} e^{4} - 40 \, B b c d x^{3} e^{3} - 20 \, A c^{2} d x^{3} e^{3} + 60 \, B b c d^{2} x^{2} e^{2} + 30 \, A c^{2} d^{2} x^{2} e^{2} - 120 \, B b c d^{3} x e - 60 \, A c^{2} d^{3} x e + 20 \, B b^{2} x^{3} e^{4} + 40 \, B a c x^{3} e^{4} + 40 \, A b c x^{3} e^{4} - 30 \, B b^{2} d x^{2} e^{3} - 60 \, B a c d x^{2} e^{3} - 60 \, A b c d x^{2} e^{3} + 60 \, B b^{2} d^{2} x e^{2} + 120 \, B a c d^{2} x e^{2} + 120 \, A b c d^{2} x e^{2} + 60 \, B a b x^{2} e^{4} + 30 \, A b^{2} x^{2} e^{4} + 60 \, A a c x^{2} e^{4} - 120 \, B a b d x e^{3} - 60 \, A b^{2} d x e^{3} - 120 \, A a c d x e^{3} + 60 \, B a^{2} x e^{4} + 120 \, A a b x e^{4}\right )} e^{\left (-5\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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