Optimal. Leaf size=43 \[ -\frac{a e^2+c d^2}{e^3 (d+e x)}-\frac{2 c d \log (d+e x)}{e^3}+\frac{c x}{e^2} \]
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Rubi [A] time = 0.0282407, antiderivative size = 43, 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} \[ -\frac{a e^2+c d^2}{e^3 (d+e x)}-\frac{2 c d \log (d+e x)}{e^3}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^2} \, dx &=\int \left (\frac{c}{e^2}+\frac{c d^2+a e^2}{e^2 (d+e x)^2}-\frac{2 c d}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{c x}{e^2}-\frac{c d^2+a e^2}{e^3 (d+e x)}-\frac{2 c d \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0208162, size = 39, normalized size = 0.91 \[ \frac{-\frac{a e^2+c d^2}{d+e x}-2 c d \log (d+e x)+c e x}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 50, normalized size = 1.2 \begin{align*}{\frac{cx}{{e}^{2}}}-2\,{\frac{cd\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{a}{e \left ( ex+d \right ) }}-{\frac{c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17228, size = 62, normalized size = 1.44 \begin{align*} -\frac{c d^{2} + a e^{2}}{e^{4} x + d e^{3}} + \frac{c x}{e^{2}} - \frac{2 \, c d \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 = 1.80624, size = 122, normalized size = 2.84 \begin{align*} \frac{c e^{2} x^{2} + c d e x - c d^{2} - a e^{2} - 2 \,{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.516944, size = 41, normalized size = 0.95 \begin{align*} - \frac{2 c d \log{\left (d + e x \right )}}{e^{3}} + \frac{c x}{e^{2}} - \frac{a e^{2} + c d^{2}}{d e^{3} + e^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3839, size = 88, normalized size = 2.05 \begin{align*}{\left (2 \, d e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c - \frac{a e^{\left (-1\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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