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 = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 47
Rule 51
Rule 63
Rule 208
Rule 266
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 1.06, size = 125, normalized size = 1.81 \[ \left [\frac {\sqrt {a} b^{2} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a^{2}}, \frac {\sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 78, normalized size = 1.13 \[ -\frac {b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{8 \, a^{\frac {3}{2}}} + \frac {1}{8} \, {\left (2 \, \sqrt {a x^{2} + b x} {\left (2 \, x + \frac {b}{a}\right )} + \frac {b^{2} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{a^{\frac {3}{2}}}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.39 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-a \,b^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+4 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x +2 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b \right ) x}{8 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 100, normalized size = 1.45 \[ \frac {b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, a^{\frac {3}{2}}} + \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} + \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} a - 2 \, {\left (a + \frac {b}{x}\right )} a^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 54, normalized size = 0.78 \[ \frac {x^2\,\sqrt {a+\frac {b}{x}}}{4}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,a^{3/2}}+\frac {x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.18, size = 97, normalized size = 1.41 \[ \frac {a x^{\frac {5}{2}}}{2 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} x^{\frac {3}{2}}}{4 \sqrt {\frac {a x}{b} + 1}} + \frac {b^{\frac {3}{2}} \sqrt {x}}{4 a \sqrt {\frac {a x}{b} + 1}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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