Optimal. Leaf size=50 \[ \frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {50, 63, 217, 203} \[ \frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx &=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b}\\ &=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b}\\ &=-\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 1.42 \[ \frac {a^{3/2} \sqrt {1-\frac {b x}{a}} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} (b x-a)}{b^{3/2} \sqrt {a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 93, normalized size = 1.86 \[ \left [-\frac {a \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, \sqrt {-b x + a} b \sqrt {x}}{2 \, b^{2}}, -\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \sqrt {-b x + a} b \sqrt {x}}{b^{2}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 1.40 \[ \frac {\sqrt {\left (-b x +a \right ) x}\, a \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, b^{\frac {3}{2}} \sqrt {x}}-\frac {\sqrt {-b x +a}\, \sqrt {x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 56, normalized size = 1.12 \[ -\frac {a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} - \frac {\sqrt {-b x + a} a}{{\left (b^{2} - \frac {{\left (b x - a\right )} b}{x}\right )} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 47, normalized size = 0.94 \[ \frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a-b\,x}-\sqrt {a}}\right )}{b^{3/2}}-\frac {\sqrt {x}\,\sqrt {a-b\,x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.28, size = 121, normalized size = 2.42 \[ \begin {cases} - \frac {i \sqrt {a} \sqrt {x} \sqrt {-1 + \frac {b x}{a}}}{b} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {x}}{b \sqrt {1 - \frac {b x}{a}}} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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