Optimal. Leaf size=47 \[ -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 47, 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 = {49, 65, 223,
209} \begin {gather*} -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx &=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-b \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 53, normalized size = 1.13 \begin {gather*} -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {-b} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right ) \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 = 2.85, size = 145, normalized size = 3.09 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 I \left (a^{\frac {3}{2}} \sqrt {\frac {-a+b x}{a}}+\sqrt {b} \sqrt {x} \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (-a+b x\right )-\sqrt {a} b x \sqrt {\frac {-a+b x}{a}}\right )}{\sqrt {x} \left (-a+b x\right )},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {-2 \sqrt {a}}{\sqrt {x} \sqrt {1-\frac {b x}{a}}}-2 \sqrt {b} \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]+\frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 66, normalized size = 1.40
method | result | size |
risch | \(-\frac {2 \sqrt {-b x +a}}{\sqrt {x}}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 35, normalized size = 0.74 \begin {gather*} 2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + a}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 91, normalized size = 1.94 \begin {gather*} \left [\frac {\sqrt {-b} x \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, \sqrt {-b x + a} \sqrt {x}}{x}, \frac {2 \, {\left (\sqrt {b} x \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} \sqrt {x}\right )}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.83, size = 148, normalized size = 3.15 \begin {gather*} \begin {cases} \frac {2 i \sqrt {a}}{\sqrt {x} \sqrt {-1 + \frac {b x}{a}}} + 2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 i b \sqrt {x}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 \sqrt {a}}{\sqrt {x} \sqrt {1 - \frac {b x}{a}}} - 2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (35) = 70\).
time = 10.30, size = 102, normalized size = 2.17 \begin {gather*} -\frac {b b^{2} \left (\frac {2 \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}}{a b-b \left (a-b x\right )}+\frac {2 \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{\sqrt {-b}}\right )}{\left |b\right | b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a-b\,x}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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