Optimal. Leaf size=25 \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]
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Rubi [A] time = 0.0377593, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2014} \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{a x^2+b x^3}} \, dx &=-\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0082031, size = 23, normalized size = 0.92 \[ -\frac{2 \sqrt{x^2 (a+b x)}}{a x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 1.1 \begin{align*} -2\,{\frac{\sqrt{x} \left ( bx+a \right ) }{a\sqrt{b{x}^{3}+a{x}^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{3} + a x^{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.713405, size = 49, normalized size = 1.96 \begin{align*} -\frac{2 \, \sqrt{b x^{3} + a x^{2}}}{a x^{\frac{3}{2}}} \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}{\sqrt{x} \sqrt{x^{2} \left (a + b x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26793, size = 41, normalized size = 1.64 \begin{align*} \frac{4 \, \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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