Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a-b x)} \]
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Rubi [A] time = 0.0142779, antiderivative size = 46, 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} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a-b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (-a+b x)^2} \, dx &=\frac{\sqrt{x}}{a (a-b x)}-\frac{\int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{2 a}\\ &=\frac{\sqrt{x}}{a (a-b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{\sqrt{x}}{a (a-b x)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0209973, size = 46, normalized size = 1. \[ \frac{\sqrt{x}}{a^2-a b x}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{a \left ( bx-a \right ) }\sqrt{x}}+{\frac{1}{a}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.71234, size = 277, normalized size = 6.02 \begin{align*} \left [-\frac{2 \, a b \sqrt{x} - \sqrt{a b}{\left (b x - a\right )} \log \left (\frac{b x + a + 2 \, \sqrt{a b} \sqrt{x}}{b x - a}\right )}{2 \,{\left (a^{2} b^{2} x - a^{3} b\right )}}, -\frac{a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )} \arctan \left (\frac{\sqrt{-a b}}{b \sqrt{x}}\right )}{a^{2} b^{2} x - a^{3} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.7792, size = 303, normalized size = 6.59 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{2} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \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 = 1.17384, size = 55, normalized size = 1.2 \begin{align*} -\frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} - \frac{\sqrt{x}}{{\left (b x - a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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