Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0100174, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-a x}}{\sqrt{x}} \, dx &=\sqrt{x} \sqrt{1-a x}+\frac{1}{2} \int \frac{1}{\sqrt{x} \sqrt{1-a x}} \, dx\\ &=\sqrt{x} \sqrt{1-a x}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0162579, size = 35, normalized size = 1. \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.159, size = 62, normalized size = 1.8 \begin{align*} \sqrt{x}\sqrt{-ax+1}+{\frac{1}{2}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{a}}}} \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 [A] time = 1.57807, size = 248, normalized size = 7.09 \begin{align*} \left [\frac{2 \, \sqrt{-a x + 1} a \sqrt{x} - \sqrt{-a} \log \left (-2 \, a x + 2 \, \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + 1\right )}{2 \, a}, \frac{\sqrt{-a x + 1} a \sqrt{x} - \sqrt{a} \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.95977, size = 83, normalized size = 2.37 \begin{align*} \begin{cases} \frac{i a x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{i \sqrt{x}}{\sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{for}\: \left |{a x}\right | > 1 \\\sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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