Optimal. Leaf size=47 \[ \frac{\sqrt{x}}{b (a-b x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
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Rubi [A] time = 0.0147455, antiderivative size = 47, 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 = {47, 63, 208} \[ \frac{\sqrt{x}}{b (a-b x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{(-a+b x)^2} \, dx &=\frac{\sqrt{x}}{b (a-b x)}+\frac{\int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{2 b}\\ &=\frac{\sqrt{x}}{b (a-b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{\sqrt{x}}{b (a-b x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0152502, size = 61, normalized size = 1.3 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x}+(b x-a) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2} (a-b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 40, normalized size = 0.9 \begin{align*} -{\frac{1}{b \left ( bx-a \right ) }\sqrt{x}}-{\frac{1}{b}{\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 = 1.65146, size = 277, normalized size = 5.89 \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 b^{3} x - a^{2} b^{2}\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 b^{3} x - a^{2} b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.04806, size = 311, normalized size = 6.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b^{2} \sqrt{x}} & \text{for}\: a = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a^{2}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} 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 = 1.20035, size = 54, normalized size = 1.15 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} - \frac{\sqrt{x}}{{\left (b x - a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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