Optimal. Leaf size=46 \[ \sqrt {x} \sqrt {a-b x}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, 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 \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}}+\sqrt {x} \sqrt {a-b x} \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 {a-b x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {a-b x}+\frac {1}{2} a \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=\sqrt {x} \sqrt {a-b x}+a \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {a-b x}+a \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=\sqrt {x} \sqrt {a-b x}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 53, normalized size = 1.15 \begin {gather*} \sqrt {x} \sqrt {a-b x}+\frac {a b \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 66, normalized size = 1.43
method | result | size |
default | \(\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(66\) |
risch | \(\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 52, normalized size = 1.13 \begin {gather*} -\frac {a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} + \frac {\sqrt {-b x + a} a}{{\left (b - \frac {b x - a}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 94, normalized size = 2.04 \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}, -\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} b \sqrt {x}}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.98, size = 119, normalized size = 2.59 \begin {gather*} \begin {cases} - \frac {i \sqrt {a} \sqrt {x}}{\sqrt {-1 + \frac {b x}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} + \frac {i b x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\sqrt {a} \sqrt {x} \sqrt {1 - \frac {b x}{a}} + \frac {a \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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 43, normalized size = 0.93 \begin {gather*} \sqrt {x}\,\sqrt {a-b\,x}+\frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a-b\,x}-\sqrt {a}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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