Optimal. Leaf size=40 \[ \frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.0141089, antiderivative size = 40, 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, 208} \[ \frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} (-a+b x)} \, dx &=\frac{2}{a \sqrt{x}}+\frac{b \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{a}\\ &=\frac{2}{a \sqrt{x}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0042176, size = 24, normalized size = 0.6 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x}{a}\right )}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 0.8 \begin{align*} -2\,{\frac{b}{a\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+2\,{\frac{1}{a\sqrt{x}}} \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 = 1.64952, size = 207, normalized size = 5.18 \begin{align*} \left [\frac{x \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 2 \, \sqrt{x}}{a x}, \frac{2 \,{\left (x \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) + \sqrt{x}\right )}}{a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.07702, size = 94, normalized size = 2.35 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2}{a \sqrt{x}} + \frac{\log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{\log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18499, size = 45, normalized size = 1.12 \begin{align*} \frac{2 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} + \frac{2}{a \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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