Optimal. Leaf size=69 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]
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Rubi [A] time = 0.0274541, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {266, 47, 51, 63, 208} \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x}} x \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \sqrt{a+\frac{b}{x}} x^2-\frac{1}{4} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b \sqrt{a+\frac{b}{x}} x}{4 a}+\frac{1}{2} \sqrt{a+\frac{b}{x}} x^2+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{b \sqrt{a+\frac{b}{x}} x}{4 a}+\frac{1}{2} \sqrt{a+\frac{b}{x}} x^2+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 a}\\ &=\frac{b \sqrt{a+\frac{b}{x}} x}{4 a}+\frac{1}{2} \sqrt{a+\frac{b}{x}} x^2-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0115659, size = 39, normalized size = 0.57 \[ \frac{2 b^2 \left (a+\frac{b}{x}\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b}{a x}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 96, normalized size = 1.4 \begin{align*}{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x+2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}b-{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{5}{2}}}} \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 = 1.79178, size = 293, normalized size = 4.25 \begin{align*} \left [\frac{\sqrt{a} b^{2} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt{\frac{a x + b}{x}}}{8 \, a^{2}}, \frac{\sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt{\frac{a x + b}{x}}}{4 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.76507, size = 97, normalized size = 1.41 \begin{align*} \frac{a x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} x^{\frac{3}{2}}}{4 \sqrt{\frac{a x}{b} + 1}} + \frac{b^{\frac{3}{2}} \sqrt{x}}{4 a \sqrt{\frac{a x}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16925, size = 105, normalized size = 1.52 \begin{align*} -\frac{b^{2} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, a^{\frac{3}{2}}} + \frac{1}{8} \,{\left (2 \, \sqrt{a x^{2} + b x}{\left (2 \, x + \frac{b}{a}\right )} + \frac{b^{2} \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{a^{\frac{3}{2}}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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