Optimal. Leaf size=17 \[ 2 d \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.0060926, 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} \[ 2 d \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx &=2 d \sqrt{a+b x+c x^2}\\ \end{align*}
Mathematica [A] time = 0.005941, size = 16, normalized size = 0.94 \[ 2 d \sqrt{a+x (b+c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 16, normalized size = 0.9 \begin{align*} 2\,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 = 2.27746, size = 20, normalized size = 1.18 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63262, size = 36, normalized size = 2.12 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.194262, size = 15, normalized size = 0.88 \begin{align*} 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 [A] time = 1.18096, size = 20, normalized size = 1.18 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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