Optimal. Leaf size=41 \[ \frac{\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac{c d x}{e^2}+\frac{c x^2}{2 e} \]
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Rubi [A] time = 0.0300263, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {697} \[ \frac{\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac{c d x}{e^2}+\frac{c x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{d+e x} \, dx &=\int \left (-\frac{c d}{e^2}+\frac{c x}{e}+\frac{c d^2+a e^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{c d x}{e^2}+\frac{c x^2}{2 e}+\frac{\left (c d^2+a e^2\right ) \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0112357, size = 38, normalized size = 0.93 \[ \frac{2 \left (a e^2+c d^2\right ) \log (d+e x)+c e x (e x-2 d)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 44, normalized size = 1.1 \begin{align*}{\frac{c{x}^{2}}{2\,e}}-{\frac{cdx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) a}{e}}+{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1435, size = 53, normalized size = 1.29 \begin{align*} \frac{c e x^{2} - 2 \, c d x}{2 \, e^{2}} + \frac{{\left (c d^{2} + a e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12247, size = 89, normalized size = 2.17 \begin{align*} \frac{c e^{2} x^{2} - 2 \, c d e x + 2 \,{\left (c d^{2} + a e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.354507, size = 36, normalized size = 0.88 \begin{align*} - \frac{c d x}{e^{2}} + \frac{c x^{2}}{2 e} + \frac{\left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24533, size = 53, normalized size = 1.29 \begin{align*}{\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c x^{2} e - 2 \, c d x\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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