Optimal. Leaf size=102 \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]
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Rubi [A] time = 0.102549, antiderivative size = 102, 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} \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]
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 )}{(d+e x)^2} \, dx &=\int \left (-\frac{c (4 c d-3 b e)}{e^3}+\frac{2 c^2 x}{e^2}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^2}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{c (4 c d-3 b e) x}{e^3}+\frac{c^2 x^2}{e^2}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0698493, size = 97, normalized size = 0.95 \[ \frac{\log (d+e x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+\frac{(2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}-c e x (4 c d-3 b e)+c^2 e^2 x^2}{e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 166, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}{x}^{2}}{{e}^{2}}}+3\,{\frac{bcx}{{e}^{2}}}-4\,{\frac{{c}^{2}dx}{{e}^{3}}}+2\,{\frac{\ln \left ( ex+d \right ) ac}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) bcd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{{e}^{4}}}-{\frac{ab}{e \left ( ex+d \right ) }}+2\,{\frac{acd}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{b}^{2}d}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{b{d}^{2}c}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04406, size = 158, normalized size = 1.55 \begin{align*} \frac{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}}{e^{5} x + d e^{4}} + \frac{c^{2} e x^{2} -{\left (4 \, c^{2} d - 3 \, b c e\right )} x}{e^{3}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} 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 = 1.69476, size = 359, normalized size = 3.52 \begin{align*} \frac{c^{2} e^{3} x^{3} + 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} - 3 \,{\left (c^{2} d e^{2} - b c e^{3}\right )} x^{2} -{\left (4 \, c^{2} d^{2} e - 3 \, b c d e^{2}\right )} x +{\left (6 \, c^{2} d^{3} - 6 \, b c d^{2} e +{\left (b^{2} + 2 \, a c\right )} d e^{2} +{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{5} x + d e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.912448, size = 124, normalized size = 1.22 \begin{align*} \frac{c^{2} x^{2}}{e^{2}} - \frac{a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 b c e - 4 c^{2} d\right )}{e^{3}} + \frac{\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} 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 = 1.15624, size = 239, normalized size = 2.34 \begin{align*}{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} -{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} e^{\left (-4\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{2 \, c^{2} d^{3} e^{2}}{x e + d} - \frac{3 \, b c d^{2} e^{3}}{x e + d} + \frac{b^{2} d e^{4}}{x e + d} + \frac{2 \, a c d e^{4}}{x e + d} - \frac{a b e^{5}}{x e + d}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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