Optimal. Leaf size=25 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {63, 205} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-a+b x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{b}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 58, normalized size = 2.32 \[ \left [-\frac {\sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right )}{a}, \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 19, normalized size = 0.76 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.80 \[ \frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 19, normalized size = 0.76 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 0.76 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.23, size = 54, normalized size = 2.16 \[ \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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