Optimal. Leaf size=107 \[ -\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{c x^2 (c d-b e)}{e^3}+\frac{x (c d-b e) (3 c d-b e)}{e^4}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x^3}{3 e^2} \]
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Rubi [A] time = 0.102462, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{c x^2 (c d-b e)}{e^3}+\frac{x (c d-b e) (3 c d-b e)}{e^4}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x^3}{3 e^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac{(c d-b e) (3 c d-b e)}{e^4}-\frac{2 c (c d-b e) x}{e^3}+\frac{c^2 x^2}{e^2}+\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^2}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{(c d-b e) (3 c d-b e) x}{e^4}-\frac{c (c d-b e) x^2}{e^3}+\frac{c^2 x^3}{3 e^2}-\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0968287, size = 114, normalized size = 1.07 \[ \frac{3 e x \left (b^2 e^2-4 b c d e+3 c^2 d^2\right )-6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (d+e x)-\frac{3 d^2 (c d-b e)^2}{d+e x}-3 c e^2 x^2 (c d-b e)+c^2 e^3 x^3}{3 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 164, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{bc{x}^{2}}{{e}^{2}}}-{\frac{{c}^{2}d{x}^{2}}{{e}^{3}}}+{\frac{{b}^{2}x}{{e}^{2}}}-4\,{\frac{bcdx}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}-2\,{\frac{d\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}}+6\,{\frac{{d}^{2}\ln \left ( ex+d \right ) bc}{{e}^{4}}}-4\,{\frac{{d}^{3}\ln \left ( ex+d \right ){c}^{2}}{{e}^{5}}}-{\frac{{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10502, size = 186, normalized size = 1.74 \begin{align*} -\frac{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac{c^{2} e^{2} x^{3} - 3 \,{\left (c^{2} d e - b c e^{2}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} x}{3 \, e^{4}} - \frac{2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65597, size = 416, normalized size = 3.89 \begin{align*} \frac{c^{2} e^{4} x^{4} - 3 \, c^{2} d^{4} + 6 \, b c d^{3} e - 3 \, b^{2} d^{2} e^{2} -{\left (2 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - 3 \, b c d^{3} e + b^{2} d^{2} e^{2} +{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.7384, size = 122, normalized size = 1.14 \begin{align*} \frac{c^{2} x^{3}}{3 e^{2}} - \frac{2 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (b c e - c^{2} d\right )}{e^{3}} + \frac{x \left (b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35191, size = 248, normalized size = 2.32 \begin{align*} \frac{1}{3} \,{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{2} d^{4} e^{3}}{x e + d} - \frac{2 \, b c d^{3} e^{4}}{x e + d} + \frac{b^{2} d^{2} e^{5}}{x e + d}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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