Integrand size = 11, antiderivative size = 86 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {15}{8} a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 86, 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, 283, 201, 223, 212} \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=-\frac {15}{8} a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )+x \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x} \]
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Rule 201
Rule 212
Rule 223
Rule 248
Rule 283
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \left (a+\frac {b}{x^2}\right )^{5/2} x-(5 b) \text {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{4} (15 a b) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{8} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{8} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right ) \\ & = -\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \left (\sqrt {b+a x^2} \left (2 b^2+9 a b x^2-8 a^2 x^4\right )+15 a^2 \sqrt {b} x^4 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )\right )}{8 x^3 \sqrt {b+a x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {b \left (9 a \,x^{2}+2 b \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{8 x^{3}}+\frac {\left (-\frac {15 \sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{2}}{8}+\sqrt {a \,x^{2}+b}\, a^{2}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b}}\) | \(100\) |
default | \(-\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x \left (-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{2} x^{4}+15 b^{\frac {5}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{2} x^{4}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a \,x^{2}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{2} b \,x^{4}-15 \sqrt {a \,x^{2}+b}\, a^{2} b^{2} x^{4}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} b \right )}{8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{2}}\) | \(144\) |
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Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.00 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=\left [\frac {15 \, a^{2} \sqrt {b} x^{3} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, x^{3}}, \frac {15 \, a^{2} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, x^{3}}\right ] \]
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Time = 2.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.36 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=\frac {a^{\frac {5}{2}} x}{\sqrt {1 + \frac {b}{a x^{2}}}} - \frac {a^{\frac {3}{2}} b}{8 x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {11 \sqrt {a} b^{2}}{8 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{8} - \frac {b^{3}}{4 \sqrt {a} x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} \]
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Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.51 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=\sqrt {a + \frac {b}{x^{2}}} a^{2} x + \frac {15}{16} \, a^{2} \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) - \frac {9 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} b x^{3} - 7 \, \sqrt {a + \frac {b}{x^{2}}} a^{2} b^{2} x}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} b x^{2} + b^{2}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.12 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=\frac {\frac {15 \, a^{3} b \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + 8 \, \sqrt {a x^{2} + b} a^{3} \mathrm {sgn}\left (x\right ) - \frac {9 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{3} b \mathrm {sgn}\left (x\right ) - 7 \, \sqrt {a x^{2} + b} a^{3} b^{2} \mathrm {sgn}\left (x\right )}{a^{2} x^{4}}}{8 \, a} \]
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Time = 6.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42 \[ \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx=\frac {x\,{\left (a\,x^2+b\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b}{a\,x^2}\right )}{{\left (\frac {b}{a}+x^2\right )}^{5/2}} \]
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