Optimal. Leaf size=35 \[ \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, 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 = {52, 56, 222}
\begin {gather*} \frac {\text {ArcSin}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {1-a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {1-a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1-a x}} \, dx\\ &=\sqrt {x} \sqrt {1-a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1-a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 52, normalized size = 1.49 \begin {gather*} \sqrt {x} \sqrt {1-a x}+\frac {a \log \left (-\sqrt {-a} \sqrt {x}+\sqrt {1-a x}\right )}{(-a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs.
\(2(25)=50\).
time = 0.09, size = 62, normalized size = 1.77
method | result | size |
meijerg | \(\frac {\sqrt {-a}\, \left (-2 \sqrt {\pi }\, \sqrt {x}\, \sqrt {-a}\, \sqrt {-a x +1}-\frac {2 \sqrt {\pi }\, \sqrt {-a}\, \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{\sqrt {a}}\right )}{2 \sqrt {\pi }\, a}\) | \(57\) |
default | \(\sqrt {x}\, \sqrt {-a x +1}+\frac {\sqrt {\left (-a x +1\right ) x}\, \arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right )}{2 \sqrt {-a x +1}\, \sqrt {x}\, \sqrt {a}}\) | \(62\) |
risch | \(-\frac {\sqrt {x}\, \left (a x -1\right ) \sqrt {\left (-a x +1\right ) x}}{\sqrt {-x \left (a x -1\right )}\, \sqrt {-a x +1}}+\frac {\sqrt {\left (-a x +1\right ) x}\, \arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right )}{2 \sqrt {-a x +1}\, \sqrt {x}\, \sqrt {a}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 48, normalized size = 1.37 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {-a x + 1}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {a}} + \frac {\sqrt {-a x + 1}}{{\left (a - \frac {a x - 1}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.17, size = 92, normalized size = 2.63 \begin {gather*} \left [\frac {2 \, \sqrt {-a x + 1} a \sqrt {x} - \sqrt {-a} \log \left (-2 \, a x + 2 \, \sqrt {-a x + 1} \sqrt {-a} \sqrt {x} + 1\right )}{2 \, a}, \frac {\sqrt {-a x + 1} a \sqrt {x} - \sqrt {a} \arctan \left (\frac {\sqrt {-a x + 1}}{\sqrt {a} \sqrt {x}}\right )}{a}\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 = 0.79, size = 82, normalized size = 2.34 \begin {gather*} \begin {cases} i \sqrt {x} \sqrt {a x - 1} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {a x^{\frac {3}{2}}}{\sqrt {- a x + 1}} + \frac {\sqrt {x}}{\sqrt {- a x + 1}} + \frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} & \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 = 2.99, size = 38, normalized size = 1.09 \begin {gather*} \sqrt {x}\,\sqrt {1-a\,x}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {1-a\,x}-1}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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