Optimal. Leaf size=42 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \]
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Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 205} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x (-a+b x)^{3/2}} \, dx &=-\frac {2}{a \sqrt {-a+b x}}-\frac {\int \frac {1}{x \sqrt {-a+b x}} \, dx}{a}\\ &=-\frac {2}{a \sqrt {-a+b x}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a b}\\ &=-\frac {2}{a \sqrt {-a+b x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.79 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-\frac {b x}{a}\right )}{a \sqrt {b x-a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 124, normalized size = 2.95 \[ \left [-\frac {{\left (b x - a\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a} a}{a^{2} b x - a^{3}}, -\frac {2 \, {\left ({\left (b x - a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \sqrt {b x - a} a\right )}}{a^{2} b x - a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 34, normalized size = 0.81 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.83 \[ -\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{\sqrt {b x -a}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 34, normalized size = 0.81 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 34, normalized size = 0.81 \[ -\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a\,\sqrt {b\,x-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.20, size = 478, normalized size = 11.38 \[ \begin {cases} \frac {2 i a^{3} \sqrt {-1 + \frac {b x}{a}}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {2 a^{3} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {2 i a^{3} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} - \frac {2 i a^{2} b x \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{i a^{\frac {9}{2}} - i a^{\frac {7}{2}} b x} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {2 a^{3} \sqrt {1 - \frac {b x}{a}}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {i \pi a^{3}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} - \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {2 a^{2} b x \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} + \frac {i \pi a^{2} b x}{- i a^{\frac {9}{2}} + i a^{\frac {7}{2}} b x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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