Optimal. Leaf size=17 \[ -\frac{1}{2 c e (d+e x)^2} \]
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Rubi [A] time = 0.0047097, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{1}{2 c e (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx &=\int \frac{1}{c (d+e x)^3} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^3} \, dx}{c}\\ &=-\frac{1}{2 c e (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0020022, size = 17, normalized size = 1. \[ -\frac{1}{2 c e (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,ce \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08266, size = 36, normalized size = 2.12 \begin{align*} -\frac{1}{2 \,{\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09091, size = 57, normalized size = 3.35 \begin{align*} -\frac{1}{2 \,{\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.506623, size = 31, normalized size = 1.82 \begin{align*} - \frac{1}{2 c d^{2} e + 4 c d e^{2} x + 2 c e^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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