Integrand size = 15, antiderivative size = 53 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {269, 52, 65, 211} \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}-\frac {2 b \sqrt {x}}{a^2}+\frac {2 x^{3/2}}{3 a} \]
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Rule 52
Rule 65
Rule 211
Rule 269
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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*}
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\frac {2 \sqrt {x} (-3 b+a x)}{3 a^2}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {2 \left (a x -3 b \right ) \sqrt {x}}{3 a^{2}}+\frac {2 b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(41\) |
derivativedivides | \(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-2 b \sqrt {x}}{a^{2}}+\frac {2 b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(44\) |
default | \(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-2 b \sqrt {x}}{a^{2}}+\frac {2 b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(44\) |
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Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\left [\frac {3 \, b \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (a x - 3 \, b\right )} \sqrt {x}}{3 \, a^{2}}, \frac {2 \, {\left (3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (a x - 3 \, b\right )} \sqrt {x}\right )}}{3 \, a^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a} - \frac {2 b \sqrt {x}}{a^{2}} + \frac {b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{a^{3} \sqrt {- \frac {b}{a}}} - \frac {b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{a^{3} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\frac {2 \, {\left (a - \frac {3 \, b}{x}\right )} x^{\frac {3}{2}}}{3 \, a^{2}} - \frac {2 \, b^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (a^{2} x^{\frac {3}{2}} - 3 \, a b \sqrt {x}\right )}}{3 \, a^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x}}{a+\frac {b}{x}} \, dx=\frac {2\,x^{3/2}}{3\,a}-\frac {2\,b\,\sqrt {x}}{a^2}+\frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]
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