Integrand size = 17, antiderivative size = 94 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=-\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 283, 201, 223, 212} \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )-\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {2}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{5/2}+\frac {10}{3} b \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2} \]
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Rule 201
Rule 212
Rule 223
Rule 283
Rule 344
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{x^4} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\frac {1}{3} (10 b) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (10 b^2\right ) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \\ \end{align*}
Time = 6.90 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {\sqrt {b+a x} \left (-3 b^2+14 a b x+2 a^2 x^2\right )}{3 x}-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )\right )}{\sqrt {b+a x}} \]
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Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {b^{2} \sqrt {\frac {a x +b}{x}}}{\sqrt {x}}+\frac {a \left (\frac {4 \left (a x +b \right )^{\frac {3}{2}}}{3}+8 b \sqrt {a x +b}-10 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{2 \sqrt {a x +b}}\) | \(82\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-2 a^{2} x^{2} \sqrt {b}\, \sqrt {a x +b}+15 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a \,b^{2} x -14 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+3 b^{\frac {5}{2}} \sqrt {a x +b}\right )}{3 \sqrt {x}\, \sqrt {a x +b}\, \sqrt {b}}\) | \(91\) |
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Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.64 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\left [\frac {15 \, a b^{\frac {3}{2}} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{6 \, x}, \frac {15 \, a \sqrt {-b} b x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, x}\right ] \]
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Time = 7.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {2 a^{2} \sqrt {b} x \sqrt {\frac {a x}{b} + 1}}{3} + \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{3} + \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a x}{b} \right )}}{2} - 5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )} - \frac {b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{x} \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {2}{3} \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a x^{\frac {3}{2}} + \frac {5}{2} \, a b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 4 \, \sqrt {a + \frac {b}{x}} a b \sqrt {x} - \frac {\sqrt {a + \frac {b}{x}} a b^{2} \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]
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Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2} + 12 \, \sqrt {a x + b} a^{2} b - \frac {3 \, \sqrt {a x + b} a b^{2}}{x}}{3 \, a} \]
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Timed out. \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\int \sqrt {x}\,{\left (a+\frac {b}{x}\right )}^{5/2} \,d x \]
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