Optimal. Leaf size=57 \[ -\frac {(b x-a)^{3/2}}{x}+3 b \sqrt {b x-a}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
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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 = {47, 50, 63, 205} \[ -\frac {(b x-a)^{3/2}}{x}+3 b \sqrt {b x-a}-3 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
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) \operatorname {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 [C] time = 0.01, size = 36, normalized size = 0.63 \[ \frac {2 b (b x-a)^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};1-\frac {b x}{a}\right )}{5 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 105, normalized size = 1.84 \[ \left [\frac {3 \, \sqrt {-a} b x \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (2 \, b x + a\right )} \sqrt {b x - a}}{2 \, x}, -\frac {3 \, \sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) - {\left (2 \, b x + a\right )} \sqrt {b x - a}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 58, normalized size = 1.02 \[ -\frac {3 \, \sqrt {a} b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) - 2 \, \sqrt {b x - a} b^{2} - \frac {\sqrt {b x - a} a b}{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.84 \[ -3 \sqrt {a}\, b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )+2 \sqrt {b x -a}\, b +\frac {\sqrt {b x -a}\, a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 47, normalized size = 0.82 \[ -3 \, \sqrt {a} b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} b + \frac {\sqrt {b x - a} a}{x} \]
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 \[ 2\,b\,\sqrt {b\,x-a}+\frac {a\,\sqrt {b\,x-a}}{x}-3\,\sqrt {a}\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.84, size = 197, normalized size = 3.46 \[ \begin {cases} - 3 i \sqrt {a} b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {i a^{2}}{\sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i a \sqrt {b}}{\sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {2 i b^{\frac {3}{2}} \sqrt {x}}{\sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\3 \sqrt {a} b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {a^{2}}{\sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {a \sqrt {b}}{\sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {2 b^{\frac {3}{2}} \sqrt {x}}{\sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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