Optimal. Leaf size=93 \[ \frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{d x (c d-b e)^2}{e^4}+\frac{c^2 x^4}{4 e} \]
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Rubi [A] time = 0.0815552, antiderivative size = 93, 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} \[ \frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{d x (c d-b e)^2}{e^4}+\frac{c^2 x^4}{4 e} \]
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} \, dx &=\int \left (-\frac{d (c d-b e)^2}{e^4}+\frac{(-c d+b e)^2 x}{e^3}-\frac{c (c d-2 b e) x^2}{e^2}+\frac{c^2 x^3}{e}+\frac{d^2 (c d-b e)^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{d (c d-b e)^2 x}{e^4}+\frac{(c d-b e)^2 x^2}{2 e^3}-\frac{c (c d-2 b e) x^3}{3 e^2}+\frac{c^2 x^4}{4 e}+\frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0420754, size = 106, normalized size = 1.14 \[ \frac{\left (b^2 d^2 e^2-2 b c d^3 e+c^2 d^4\right ) \log (d+e x)}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{x^2 (b e-c d)^2}{2 e^3}-\frac{d x (c d-b e)^2}{e^4}+\frac{c^2 x^4}{4 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 152, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}{x}^{4}}{4\,e}}+{\frac{2\,bc{x}^{3}}{3\,e}}-{\frac{{c}^{2}d{x}^{3}}{3\,{e}^{2}}}+{\frac{{b}^{2}{x}^{2}}{2\,e}}-{\frac{bc{x}^{2}d}{{e}^{2}}}+{\frac{{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{{b}^{2}dx}{{e}^{2}}}+2\,{\frac{bc{d}^{2}x}{{e}^{3}}}-{\frac{{c}^{2}{d}^{3}x}{{e}^{4}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}}-2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) bc}{{e}^{4}}}+{\frac{{d}^{4}\ln \left ( ex+d \right ){c}^{2}}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15742, size = 177, normalized size = 1.9 \begin{align*} \frac{3 \, c^{2} e^{3} x^{4} - 4 \,{\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} x}{12 \, e^{4}} + \frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} 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 = 1.53304, size = 279, normalized size = 3. \begin{align*} \frac{3 \, c^{2} e^{4} x^{4} - 4 \,{\left (c^{2} d e^{3} - 2 \, b c e^{4}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x + 12 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.958715, size = 112, normalized size = 1.2 \begin{align*} \frac{c^{2} x^{4}}{4 e} + \frac{d^{2} \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{x^{3} \left (2 b c e - c^{2} d\right )}{3 e^{2}} + \frac{x^{2} \left (b^{2} e^{2} - 2 b c d e + c^{2} d^{2}\right )}{2 e^{3}} - \frac{x \left (b^{2} d e^{2} - 2 b c d^{2} e + c^{2} d^{3}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28223, size = 181, normalized size = 1.95 \begin{align*}{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 8 \, b c x^{3} e^{3} - 12 \, b c d x^{2} e^{2} + 24 \, b c d^{2} x e + 6 \, b^{2} x^{2} e^{3} - 12 \, b^{2} d x e^{2}\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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