Optimal. Leaf size=53 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]
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Rubi [A] time = 0.0181793, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{a+\frac{b}{x}} \, dx &=\int \frac{x^{3/2}}{b+a x} \, dx\\ &=\frac{2 x^{3/2}}{3 a}-\frac{b \int \frac{\sqrt{x}}{b+a x} \, dx}{a}\\ &=-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a}+\frac{b^2 \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{a^2}\\ &=-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a}+\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.017519, size = 49, normalized size = 0.92 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}+\frac{2 \sqrt{x} (a x-3 b)}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 43, normalized size = 0.8 \begin{align*}{\frac{2}{3\,a}{x}^{{\frac{3}{2}}}}-2\,{\frac{b\sqrt{x}}{{a}^{2}}}+2\,{\frac{{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 1.72931, size = 244, normalized size = 4.6 \begin{align*} \left [\frac{3 \, b \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (a x - 3 \, b\right )} \sqrt{x}}{3 \, a^{2}}, \frac{2 \,{\left (3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) +{\left (a x - 3 \, b\right )} \sqrt{x}\right )}}{3 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.32869, size = 105, normalized size = 1.98 \begin{align*} \begin{cases} \frac{2 x^{\frac{3}{2}}}{3 a} - \frac{2 b \sqrt{x}}{a^{2}} - \frac{i b^{\frac{3}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} + \frac{i b^{\frac{3}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{5}{2}}}{5 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10151, size = 61, normalized size = 1.15 \begin{align*} \frac{2 \, b^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{2 \,{\left (a^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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