Optimal. Leaf size=52 \[ -\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0299414, antiderivative size = 52, 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} \[ -\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^4} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^4}-\frac{2 c d}{e^2 (d+e x)^3}+\frac{c}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac{c d^2+a e^2}{3 e^3 (d+e x)^3}+\frac{c d}{e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0134262, size = 39, normalized size = 0.75 \[ -\frac{a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 51, normalized size = 1. \begin{align*} -{\frac{a{e}^{2}+c{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }}+{\frac{cd}{{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 = 1.18936, size = 85, normalized size = 1.63 \begin{align*} -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91301, size = 130, normalized size = 2.5 \begin{align*} -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.77414, size = 66, normalized size = 1.27 \begin{align*} - \frac{a e^{2} + c d^{2} + 3 c d e x + 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28112, size = 50, normalized size = 0.96 \begin{align*} -\frac{{\left (3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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