Optimal. Leaf size=43 \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {242, 51, 63, 208} \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 242
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 98, normalized size = 2.28 \[ \left [\frac {2 \, a x \sqrt {\frac {a x + b}{x}} + \sqrt {a} b \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a^{2}}, \frac {a x \sqrt {\frac {a x + b}{x}} + \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 71, normalized size = 1.65 \[ -\frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{2 \, a^{\frac {3}{2}}} + \frac {b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {3}{2}} \mathrm {sgn}\relax (x)} + \frac {\sqrt {a x^{2} + b x}}{a \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 1.65 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-b \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 67, normalized size = 1.56 \[ \frac {\sqrt {a + \frac {b}{x}} b}{{\left (a + \frac {b}{x}\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 66, normalized size = 1.53 \[ \frac {2\,x\,\left (\frac {3\,\sqrt {b}\,\sqrt {b+a\,x}}{2\,a\,x}+\frac {b^{3/2}\,\mathrm {asin}\left (\frac {\sqrt {a}\,\sqrt {x}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,3{}\mathrm {i}}{2\,a^{3/2}\,x^{3/2}}\right )\,\sqrt {\frac {a\,x}{b}+1}}{3\,\sqrt {a+\frac {b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.09, size = 44, normalized size = 1.02 \[ \frac {\sqrt {b} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{a} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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