Optimal. Leaf size=104 \[ \frac{x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{c x^2 (2 c d-3 b e)}{2 e^2}+\frac{2 c^2 x^3}{3 e} \]
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Rubi [A] time = 0.111707, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{c x^2 (2 c d-3 b e)}{2 e^2}+\frac{2 c^2 x^3}{3 e} \]
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 )}{d+e x} \, dx &=\int \left (\frac{2 c^2 d^2+b^2 e^2-c e (3 b d-2 a e)}{e^3}-\frac{c (2 c d-3 b e) x}{e^2}+\frac{2 c^2 x^2}{e}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{\left (2 c^2 d^2+b^2 e^2-c e (3 b d-2 a e)\right ) x}{e^3}-\frac{c (2 c d-3 b e) x^2}{2 e^2}+\frac{2 c^2 x^3}{3 e}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.03969, size = 95, normalized size = 0.91 \[ \frac{e x \left (3 c e (4 a e-6 b d+3 b e x)+6 b^2 e^2+2 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 146, normalized size = 1.4 \begin{align*}{\frac{2\,{c}^{2}{x}^{3}}{3\,e}}+{\frac{3\,b{x}^{2}c}{2\,e}}-{\frac{{x}^{2}{c}^{2}d}{{e}^{2}}}+2\,{\frac{acx}{e}}+{\frac{{b}^{2}x}{e}}-3\,{\frac{bcdx}{{e}^{2}}}+2\,{\frac{{c}^{2}{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) ab}{e}}-2\,{\frac{\ln \left ( ex+d \right ) acd}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ){b}^{2}d}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) bc{d}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{3}}{{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999469, size = 157, normalized size = 1.51 \begin{align*} \frac{4 \, c^{2} e^{2} x^{3} - 3 \,{\left (2 \, c^{2} d e - 3 \, b c e^{2}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{2} - 3 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68656, size = 254, normalized size = 2.44 \begin{align*} \frac{4 \, c^{2} e^{3} x^{3} - 3 \,{\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.551199, size = 104, normalized size = 1. \begin{align*} \frac{2 c^{2} x^{3}}{3 e} + \frac{x^{2} \left (3 b c e - 2 c^{2} d\right )}{2 e^{2}} + \frac{x \left (2 a c e^{2} + b^{2} e^{2} - 3 b c d e + 2 c^{2} d^{2}\right )}{e^{3}} + \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18013, size = 159, normalized size = 1.53 \begin{align*} -{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (4 \, c^{2} x^{3} e^{2} - 6 \, c^{2} d x^{2} e + 12 \, c^{2} d^{2} x + 9 \, b c x^{2} e^{2} - 18 \, b c d x e + 6 \, b^{2} x e^{2} + 12 \, a c x e^{2}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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