Optimal. Leaf size=51 \[ -\frac{a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.0324775, antiderivative size = 51, 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} \[ -\frac{a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^3} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^3}-\frac{2 c d}{e^2 (d+e x)^2}+\frac{c}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{c d^2+a e^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.015273, size = 48, normalized size = 0.94 \[ \frac{-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 56, normalized size = 1.1 \begin{align*}{\frac{c\ln \left ( ex+d \right ) }{{e}^{3}}}+2\,{\frac{cd}{{e}^{3} \left ( ex+d \right ) }}-{\frac{a}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16519, size = 77, normalized size = 1.51 \begin{align*} \frac{4 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{c \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 = 1.78359, size = 157, normalized size = 3.08 \begin{align*} \frac{4 \, c d e x + 3 \, c d^{2} - a e^{2} + 2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.613881, size = 56, normalized size = 1.1 \begin{align*} \frac{c \log{\left (d + e x \right )}}{e^{3}} + \frac{- a e^{2} + 3 c d^{2} + 4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41966, size = 62, normalized size = 1.22 \begin{align*} c e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (4 \, c d x +{\left (3 \, c d^{2} - a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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