Optimal. Leaf size=223 \[ \frac{c x^2 \left (-c e (5 b d-2 a e)+2 b^2 e^2+3 c^2 d^2\right )}{e^4}-\frac{x \left (-c^2 d e (15 b d-8 a e)+2 b c e^2 (4 b d-3 a e)-b^3 e^3+8 c^3 d^3\right )}{e^5}+\frac{2 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac{c^2 x^3 (4 c d-5 b e)}{3 e^3}+\frac{c^3 x^4}{2 e^2} \]
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Rubi [A] time = 0.289959, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{c x^2 \left (-c e (5 b d-2 a e)+2 b^2 e^2+3 c^2 d^2\right )}{e^4}-\frac{x \left (-c^2 d e (15 b d-8 a e)+2 b c e^2 (4 b d-3 a e)-b^3 e^3+8 c^3 d^3\right )}{e^5}+\frac{2 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac{c^2 x^3 (4 c d-5 b e)}{3 e^3}+\frac{c^3 x^4}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac{-8 c^3 d^3+b^3 e^3+c^2 d e (15 b d-8 a e)-2 b c e^2 (4 b d-3 a e)}{e^5}+\frac{2 c \left (3 c^2 d^2+2 b^2 e^2-c e (5 b d-2 a e)\right ) x}{e^4}-\frac{c^2 (4 c d-5 b e) x^2}{e^3}+\frac{2 c^3 x^3}{e^2}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^2}+\frac{2 \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^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (8 c^3 d^3-b^3 e^3-c^2 d e (15 b d-8 a e)+2 b c e^2 (4 b d-3 a e)\right ) x}{e^5}+\frac{c \left (3 c^2 d^2+2 b^2 e^2-c e (5 b d-2 a e)\right ) x^2}{e^4}-\frac{c^2 (4 c d-5 b e) x^3}{3 e^3}+\frac{c^3 x^4}{2 e^2}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)}+\frac{2 \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 ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.200251, size = 241, normalized size = 1.08 \[ \frac{12 \log (d+e x) \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+6 c e^2 x^2 \left (c e (2 a e-5 b d)+2 b^2 e^2+3 c^2 d^2\right )+6 e x \left (c^2 d e (15 b d-8 a e)+2 b c e^2 (3 a e-4 b d)+b^3 e^3-8 c^3 d^3\right )+\frac{6 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}-2 c^2 e^3 x^3 (4 c d-5 b e)+3 c^3 e^4 x^4}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 444, normalized size = 2. \begin{align*} 2\,{\frac{{b}^{2}da}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{d{a}^{2}c}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{c}^{3}{x}^{4}}{2\,{e}^{2}}}-12\,{\frac{\ln \left ( ex+d \right ) abcd}{{e}^{3}}}-6\,{\frac{abc{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-8\,{\frac{{c}^{3}{d}^{3}x}{{e}^{5}}}+3\,{\frac{{x}^{2}{c}^{3}{d}^{2}}{{e}^{4}}}+2\,{\frac{{b}^{2}{x}^{2}c}{{e}^{2}}}+{\frac{5\,b{x}^{3}{c}^{2}}{3\,{e}^{2}}}-{\frac{4\,{x}^{3}{c}^{3}d}{3\,{e}^{3}}}+2\,{\frac{a{x}^{2}{c}^{2}}{{e}^{2}}}+2\,{\frac{{c}^{3}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+10\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{4}}{{e}^{6}}}+2\,{\frac{\ln \left ( ex+d \right ) c{a}^{2}}{{e}^{2}}}+2\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ){b}^{3}d}{{e}^{3}}}-{\frac{{a}^{2}b}{e \left ( ex+d \right ) }}-{\frac{{b}^{3}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}x}{{e}^{2}}}+12\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{2}}{{e}^{4}}}-20\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{3}}{{e}^{5}}}+6\,{\frac{abcx}{{e}^{2}}}-5\,{\frac{b{x}^{2}{c}^{2}d}{{e}^{3}}}+12\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{2}}{{e}^{4}}}-8\,{\frac{{b}^{2}cdx}{{e}^{3}}}+15\,{\frac{b{d}^{2}{c}^{2}x}{{e}^{4}}}+4\,{\frac{a{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{b}^{2}{d}^{3}c}{{e}^{4} \left ( ex+d \right ) }}-5\,{\frac{b{d}^{4}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-8\,{\frac{a{c}^{2}dx}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02214, size = 419, normalized size = 1.88 \begin{align*} \frac{2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}}{e^{7} x + d e^{6}} + \frac{3 \, c^{3} e^{3} x^{4} - 2 \,{\left (4 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x^{3} + 6 \,{\left (3 \, c^{3} d^{2} e - 5 \, b c^{2} d e^{2} + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 6 \,{\left (8 \, c^{3} d^{3} - 15 \, b c^{2} d^{2} e + 8 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{6 \, e^{5}} + \frac{2 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\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 = 1.49618, size = 921, normalized size = 4.13 \begin{align*} \frac{3 \, c^{3} e^{5} x^{5} + 12 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, a^{2} b e^{5} + 24 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 6 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 12 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 5 \,{\left (c^{3} d e^{4} - 2 \, b c^{2} e^{5}\right )} x^{4} + 2 \,{\left (5 \, c^{3} d^{2} e^{3} - 10 \, b c^{2} d e^{4} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 6 \,{\left (5 \, c^{3} d^{3} e^{2} - 10 \, b c^{2} d^{2} e^{3} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 6 \,{\left (8 \, c^{3} d^{4} e - 15 \, b c^{2} d^{3} e^{2} + 8 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4}\right )} x + 12 \,{\left (5 \, c^{3} d^{5} - 10 \, b c^{2} d^{4} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} +{\left (a b^{2} + a^{2} c\right )} d e^{4} +{\left (5 \, c^{3} d^{4} e - 10 \, b c^{2} d^{3} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} +{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\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 = 2.18455, size = 318, normalized size = 1.43 \begin{align*} \frac{c^{3} x^{4}}{2 e^{2}} - \frac{a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (5 b c^{2} e - 4 c^{3} d\right )}{3 e^{3}} + \frac{x^{2} \left (2 a c^{2} e^{2} + 2 b^{2} c e^{2} - 5 b c^{2} d e + 3 c^{3} d^{2}\right )}{e^{4}} + \frac{x \left (6 a b c e^{3} - 8 a c^{2} d e^{2} + b^{3} e^{3} - 8 b^{2} c d e^{2} + 15 b c^{2} d^{2} e - 8 c^{3} d^{3}\right )}{e^{5}} + \frac{2 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} 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 [A] time = 1.34768, size = 567, normalized size = 2.54 \begin{align*} \frac{1}{6} \,{\left (3 \, c^{3} - \frac{10 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{12 \,{\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{6 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b 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 )} - 2 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c 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{2 \, c^{3} d^{5} e^{4}}{x e + d} - \frac{5 \, b c^{2} d^{4} e^{5}}{x e + d} + \frac{4 \, b^{2} c d^{3} e^{6}}{x e + d} + \frac{4 \, a c^{2} d^{3} e^{6}}{x e + d} - \frac{b^{3} d^{2} e^{7}}{x e + d} - \frac{6 \, a b c d^{2} e^{7}}{x e + d} + \frac{2 \, a b^{2} d e^{8}}{x e + d} + \frac{2 \, a^{2} c d e^{8}}{x e + d} - \frac{a^{2} b 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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