Optimal. Leaf size=29 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0125623, antiderivative size = 29, 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 = {63, 217, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.014671, size = 52, normalized size = 1.79 \[ \frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 51, normalized size = 1.8 \begin{align*}{\sqrt{x \left ( -bx+a \right ) }\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{b}}}} \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 = 2.13147, size = 165, normalized size = 5.69 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right )}{b}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35913, size = 56, normalized size = 1.93 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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