Integrand size = 13, antiderivative size = 66 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 43, 65, 214} \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}+\frac {3}{4} b x \sqrt {a+\frac {b}{x}} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{4} (3 b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{8} \left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2-\frac {1}{4} (3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right ) \\ & = \frac {3}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {1}{4} \left (\sqrt {a+\frac {b}{x}} x (5 b+2 a x)+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26
method | result | size |
risch | \(\frac {\left (2 a x +5 b \right ) x \sqrt {\frac {a x +b}{x}}}{4}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{8 \sqrt {a}\, \left (a x +b \right )}\) | \(83\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x}\, x +10 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b +3 b^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \right )}{8 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}}}\) | \(96\) |
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.97 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a}\right ] \]
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Time = 1.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{2} + \frac {5 b^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{4} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \]
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Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.48 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} - 2 \, {\left (a + \frac {b}{x}\right )} a + a^{2}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=-\frac {3 \, b^{2} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {1}{4} \, \sqrt {a x^{2} + b x} {\left (2 \, a x \mathrm {sgn}\left (x\right ) + 5 \, b \mathrm {sgn}\left (x\right )\right )} \]
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Time = 5.82 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {5\,x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4}+\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,\sqrt {a}}-\frac {3\,a\,x^2\,\sqrt {a+\frac {b}{x}}}{4} \]
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