Integrand size = 17, antiderivative size = 106 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 106, 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, 285, 327, 223, 212} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{5/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}-\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}} \]
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 344
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int x^4 \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {1}{3} a \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{8 b^2} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^2} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{5/2}} \\ \end{align*}
Time = 10.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (\frac {\sqrt {b} \left (-8 b^2-2 a b x+3 a^2 x^2\right )}{x^{5/2}}-\frac {3 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {1+\frac {b}{a x}}}\right )}{24 b^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {\left (3 a^{2} x^{2}-2 a b x -8 b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{24 x^{\frac {5}{2}} b^{2}}-\frac {a^{3} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{8 b^{\frac {5}{2}} \sqrt {a x +b}}\) | \(81\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{3} x^{3}-3 a^{2} x^{2} \sqrt {b}\, \sqrt {a x +b}+2 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+8 b^{\frac {5}{2}} \sqrt {a x +b}\right )}{24 x^{\frac {5}{2}} b^{\frac {5}{2}} \sqrt {a x +b}}\) | \(92\) |
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Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\left [\frac {3 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a^{2} b x^{2} - 2 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b^{3} x^{3}}, \frac {3 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a^{2} b x^{2} - 2 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b^{3} x^{3}}\right ] \]
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Time = 10.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {a^{\frac {5}{2}}}{8 b^{2} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} + \frac {a^{\frac {3}{2}}}{24 b x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {5 \sqrt {a}}{12 x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 b^{\frac {5}{2}}} - \frac {b}{3 \sqrt {a} x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {5}{2}}} + \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} - 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} - 3 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} b^{2} x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b^{3} x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{4} x - b^{5}\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {{\left (\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} - 8 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b - 3 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} b^{2} x^{3}}\right )} \mathrm {sgn}\left (x\right )}{24 \, a} \]
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Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \,d x \]
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