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 = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 217
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*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.79 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 38, normalized size = 1.31 \begin {gather*} \frac {2 \sqrt {-b} \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 57, normalized size = 1.97 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 51, normalized size = 1.76 \begin {gather*} \frac {\sqrt {\left (-b x +a \right ) x}\, \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{\sqrt {-b x +a}\, \sqrt {b}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 21, normalized size = 0.72 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 27, normalized size = 0.93 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {a-b\,x}-\sqrt {a}}{\sqrt {b}\,\sqrt {x}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 54, normalized size = 1.86 \begin {gather*} \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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