Integrand size = 11, antiderivative size = 79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {248, 44, 53, 65, 214} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 b}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 248
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3} \\ & = \frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^3} \\ & = \frac {5 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^2+20 a b x+3 a^2 x^2\right )}{3 a^3 (b+a x)^2}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(65)=130\).
Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.99
method | result | size |
risch | \(\frac {a x +b}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {5 b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{2 a^{\frac {7}{2}}}-\frac {2 b^{2} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{5} \left (x +\frac {b}{a}\right )^{2}}+\frac {14 b \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 a^{4} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(157\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-30 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} x^{3}+15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,x^{3}+24 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} x -90 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b \,x^{2}+45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} x^{2}+20 b \,a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-90 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2} x +45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} x -30 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{3}+15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4}\right )}{6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{3}}\) | \(271\) |
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Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (3 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}, \frac {15 \, {\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (66) = 132\).
Time = 2.63 (sec) , antiderivative size = 774, normalized size of antiderivative = 9.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {6 a^{17} x^{4} \sqrt {1 + \frac {b}{a x}}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {46 a^{16} b x^{3} \sqrt {1 + \frac {b}{a x}}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{16} b x^{3} \log {\left (\frac {b}{a x} \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{16} b x^{3} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {70 a^{15} b^{2} x^{2} \sqrt {1 + \frac {b}{a x}}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{15} b^{2} x^{2} \log {\left (\frac {b}{a x} \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{15} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {30 a^{14} b^{3} x \sqrt {1 + \frac {b}{a x}}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{14} b^{3} x \log {\left (\frac {b}{a x} \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{14} b^{3} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{13} b^{4} \log {\left (\frac {b}{a x} \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{13} b^{4} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{6 a^{\frac {39}{2}} x^{3} + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x + 6 a^{\frac {33}{2}} b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b}{3 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}\right )}} + \frac {5 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (65) = 130\).
Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {{\left (15 \, b \log \left ({\left | b \right |}\right ) + 28 \, b\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}}} + \frac {5 \, b \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} + \frac {\sqrt {a x^{2} + b x}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} + 7 \, b^{4}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
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Time = 5.82 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2\,x\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {7}{2};\ \frac {9}{2};\ -\frac {a\,x}{b}\right )}{7\,{\left (a+\frac {b}{x}\right )}^{5/2}} \]
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