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 = 0.0099389, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}
Mathematica [C] time = 0.0082974, 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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Maple [A] time = 0.007, size = 35, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{{a}^{3/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) }-2\,{\frac{1}{a\sqrt{bx-a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44623, size = 266, normalized size = 6.33 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.73861, size = 439, normalized size = 10.45 \begin{align*} \begin{cases} - \frac{2 a^{3} \sqrt{-1 + \frac{b x}{a}}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{i a^{3} \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 i a^{3} \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{i a^{2} b x \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 i a^{2} b x \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 a^{2} b x \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{2 i a^{3} \sqrt{1 - \frac{b x}{a}}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{i a^{3} \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 i a^{3} \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{\pi a^{3}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{i a^{2} b x \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 i a^{2} b x \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{\pi a^{2} b x}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17981, size = 46, normalized size = 1.1 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} - \frac{2}{\sqrt{b x - a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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