Optimal. Leaf size=20 \[ -\frac{1}{2 c d (a e+c d x)^2} \]
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Rubi [A] time = 0.0102619, antiderivative size = 20, 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} \[ -\frac{1}{2 c d (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{1}{(a e+c d x)^3} \, dx\\ &=-\frac{1}{2 c d (a e+c d x)^2}\\ \end{align*}
Mathematica [A] time = 0.0035411, size = 20, normalized size = 1. \[ -\frac{1}{2 c d (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{2\,cd \left ( cdx+ae \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0685, size = 47, normalized size = 2.35 \begin{align*} -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94773, size = 70, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.450763, size = 39, normalized size = 1.95 \begin{align*} - \frac{1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37128, size = 350, normalized size = 17.5 \begin{align*} -\frac{c^{4} d^{8} x^{2} e^{2} + 2 \, c^{4} d^{9} x e + c^{4} d^{10} - 4 \, a c^{3} d^{6} x^{2} e^{4} - 8 \, a c^{3} d^{7} x e^{3} - 4 \, a c^{3} d^{8} e^{2} + 6 \, a^{2} c^{2} d^{4} x^{2} e^{6} + 12 \, a^{2} c^{2} d^{5} x e^{5} + 6 \, a^{2} c^{2} d^{6} e^{4} - 4 \, a^{3} c d^{2} x^{2} e^{8} - 8 \, a^{3} c d^{3} x e^{7} - 4 \, a^{3} c d^{4} e^{6} + a^{4} x^{2} e^{10} + 2 \, a^{4} d x e^{9} + a^{4} d^{2} e^{8}}{2 \,{\left (c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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