Optimal. Leaf size=50 \[ -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.01, 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 = {52, 65, 223,
209} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 223
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 \text {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 \text {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 = 0.06, size = 56, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.17, size = 135, normalized size = 2.70 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \sqrt {a} \left (-\sqrt {a} b \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (-a+b x\right )+a b^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {-a+b x}{a}}-b^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {\frac {-a+b x}{a}}\right )}{b^{\frac {5}{2}} \left (-a+b x\right )},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {a \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{b^{\frac {3}{2}}}-\frac {\sqrt {a} \sqrt {x} \sqrt {1-\frac {b x}{a}}}{b}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 70, normalized size = 1.40
method | result | size |
default | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(70\) |
risch | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 56, normalized size = 1.12 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 93, normalized size = 1.86 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.19, size = 121, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {i \sqrt {a} \sqrt {x}}{b \sqrt {-1 + \frac {b x}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {i x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {x} \sqrt {1 - \frac {b x}{a}}}{b} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 67, normalized size = 1.34 \begin {gather*} 2 \left (-\frac {\frac {1}{4}\cdot 2 \sqrt {x} \sqrt {a-b x}}{b}-\frac {2 a \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{4 b \sqrt {-b}}\right ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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