Optimal. Leaf size=18 \[ -\frac{1}{c d (a e+c d x)} \]
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Rubi [A] time = 0.0098234, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 32} \[ -\frac{1}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{1}{(a e+c d x)^2} \, dx\\ &=-\frac{1}{c d (a e+c d x)}\\ \end{align*}
Mathematica [A] time = 0.0038684, size = 18, normalized size = 1. \[ -\frac{1}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 19, normalized size = 1.1 \begin{align*} -{\frac{1}{cd \left ( cdx+ae \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01625, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{c^{2} d^{2} x + a c d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73763, size = 35, normalized size = 1.94 \begin{align*} -\frac{1}{c^{2} d^{2} x + a c d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.372318, size = 17, normalized size = 0.94 \begin{align*} - \frac{1}{a c d e + c^{2} d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19251, size = 147, normalized size = 8.17 \begin{align*} -\frac{c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}}{{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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