Optimal. Leaf size=57 \[ 3 b \sqrt {-a+b x}-\frac {(-a+b x)^{3/2}}{x}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 52, 65, 211}
\begin {gather*} -\frac {(b x-a)^{3/2}}{x}+3 b \sqrt {b x-a}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {(-a+b x)^{3/2}}{x^2} \, dx &=-\frac {(-a+b x)^{3/2}}{x}+\frac {1}{2} (3 b) \int \frac {\sqrt {-a+b x}}{x} \, dx\\ &=3 b \sqrt {-a+b x}-\frac {(-a+b x)^{3/2}}{x}-\frac {1}{2} (3 a b) \int \frac {1}{x \sqrt {-a+b x}} \, dx\\ &=3 b \sqrt {-a+b x}-\frac {(-a+b x)^{3/2}}{x}-(3 a) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )\\ &=3 b \sqrt {-a+b x}-\frac {(-a+b x)^{3/2}}{x}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.84 \begin {gather*} \frac {\sqrt {-a+b x} (a+2 b x)}{x}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\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 = 3.79, size = 194, normalized size = 3.40 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (a^2 \left (a-b x\right )+a b x \left (a-b x\right )-2 b^2 x^2 \left (a-b x\right )-3 \sqrt {a} b^{\frac {5}{2}} x^{\frac {5}{2}} \text {ArcCosh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ] \left (\frac {a-b x}{b x}\right )^{\frac {3}{2}}\right )}{b^{\frac {3}{2}} x^{\frac {5}{2}} \left (\frac {a-b x}{b x}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},-\frac {a^2}{\sqrt {b} x^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}-\frac {a \sqrt {b}}{\sqrt {x} \sqrt {1-\frac {a}{b x}}}+3 \sqrt {a} b \text {ArcSin}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]+\frac {2 b^{\frac {3}{2}} \sqrt {x}}{\sqrt {1-\frac {a}{b x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 54, normalized size = 0.95
method | result | size |
derivativedivides | \(2 b \left (\sqrt {b x -a}-a \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(54\) |
default | \(2 b \left (\sqrt {b x -a}-a \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(54\) |
risch | \(-\frac {a \left (-b x +a \right )}{x \sqrt {b x -a}}-3 b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) \sqrt {a}+2 b \sqrt {b x -a}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 47, normalized size = 0.82 \begin {gather*} -3 \, \sqrt {a} b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} b + \frac {\sqrt {b x - a} a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 105, normalized size = 1.84 \begin {gather*} \left [\frac {3 \, \sqrt {-a} b x \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (2 \, b x + a\right )} \sqrt {b x - a}}{2 \, x}, -\frac {3 \, \sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) - {\left (2 \, b x + a\right )} \sqrt {b x - a}}{x}\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 = 1.36, size = 197, normalized size = 3.46 \begin {gather*} \begin {cases} - 3 i \sqrt {a} b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {i a^{2}}{\sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i a \sqrt {b}}{\sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {2 i b^{\frac {3}{2}} \sqrt {x}}{\sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\3 \sqrt {a} b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {a^{2}}{\sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {a \sqrt {b}}{\sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {2 b^{\frac {3}{2}} \sqrt {x}}{\sqrt {- \frac {a}{b x} + 1}} & \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 = 75, normalized size = 1.32 \begin {gather*} \frac {2 \sqrt {-a+b x} b^{2}+\frac {\sqrt {-a+b x} b^{2} a}{-a+b x+a}-\frac {6 b^{2} a \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 47, normalized size = 0.82 \begin {gather*} 2\,b\,\sqrt {b\,x-a}+\frac {a\,\sqrt {b\,x-a}}{x}-3\,\sqrt {a}\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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