Optimal. Leaf size=17 \[ -\frac{d}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0048227, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {629} \[ -\frac{d}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int \frac{b d+2 c d x}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d}{2 \left (a+b x+c x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0064625, size = 16, normalized size = 0.94 \[ -\frac{d}{2 (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 16, normalized size = 0.9 \begin{align*} -{\frac{d}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33547, size = 20, normalized size = 1.18 \begin{align*} -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02136, size = 89, normalized size = 5.24 \begin{align*} -\frac{d}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.81857, size = 44, normalized size = 2.59 \begin{align*} - \frac{d}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22106, size = 20, normalized size = 1.18 \begin{align*} -\frac{d}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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