Optimal. Leaf size=17 \[ -\frac{2 d}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0061819, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {629} \[ -\frac{2 d}{\sqrt{a+b x+c x^2}} \]
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 )^{3/2}} \, dx &=-\frac{2 d}{\sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0069606, size = 16, normalized size = 0.94 \[ -\frac{2 d}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 16, normalized size = 0.9 \begin{align*} -2\,{\frac{d}{\sqrt{c{x}^{2}+bx+a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1367, size = 20, normalized size = 1.18 \begin{align*} -\frac{2 \, d}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.19535, size = 38, normalized size = 2.24 \begin{align*} -\frac{2 \, d}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49275, size = 17, normalized size = 1. \begin{align*} - \frac{2 d}{\sqrt{a + b x + c x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16329, size = 47, normalized size = 2.76 \begin{align*} -\frac{2 \,{\left (b^{2} d - 4 \, a c d\right )}}{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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