Optimal. Leaf size=32 \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0042317, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {613} \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0172564, size = 31, normalized size = 0.97 \[ -\frac{2 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 33, normalized size = 1. \begin{align*} 2\,{\frac{2\,cx+b}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.76706, size = 135, normalized size = 4.22 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13739, size = 55, normalized size = 1.72 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c x}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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