Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {5 a^4 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{128 b^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 285, 327, 223, 212} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\frac {5 a^4 \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{128 b^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}-\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3} \]
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 342
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 \left (a+b x^2\right )^{5/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {1}{8} (5 a) \text {Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {1}{16} \left (5 a^2\right ) \text {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {1}{64} \left (5 a^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )}{128 b} \\ & = -\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )}{128 b} \\ & = -\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{128 b^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (48 b^3+136 a b^2 x^2+118 a^2 b x^4+15 a^3 x^6\right )+\frac {15 a^4 x^8 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{384 b^{3/2} x^7} \]
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Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {\left (15 x^{6} a^{3}+118 a^{2} b \,x^{4}+136 a \,b^{2} x^{2}+48 b^{3}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{384 x^{7} b}+\frac {5 a^{4} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{128 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\) | \(109\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{4} x^{8}+15 b^{\frac {5}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{4} x^{8}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{3} x^{6}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{4} b \,x^{8}-15 \sqrt {a \,x^{2}+b}\, a^{4} b^{2} x^{8}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{2} b \,x^{4}+8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a \,b^{2} x^{2}-48 \left (a \,x^{2}+b \right )^{\frac {7}{2}} b^{3}\right )}{384 x^{3} \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{4}}\) | \(186\) |
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Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\left [\frac {15 \, a^{4} \sqrt {b} x^{7} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{768 \, b^{2} x^{7}}, -\frac {15 \, a^{4} \sqrt {-b} x^{7} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{384 \, b^{2} x^{7}}\right ] \]
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Time = 6.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=- \frac {5 a^{\frac {7}{2}}}{128 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {133 a^{\frac {5}{2}}}{384 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {127 a^{\frac {3}{2}} b}{192 x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {23 \sqrt {a} b^{2}}{48 x^{7} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{128 b^{\frac {3}{2}}} - \frac {b^{3}}{8 \sqrt {a} x^{9} \sqrt {1 + \frac {b}{a x^{2}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 \, a^{4} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{256 \, b^{\frac {3}{2}}} - \frac {15 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}} a^{4} x^{7} + 73 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{4} b x^{5} - 55 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{4} b^{2} x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} a^{4} b^{3} x}{384 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{4} b x^{8} - 4 \, {\left (a + \frac {b}{x^{2}}\right )}^{3} b^{2} x^{6} + 6 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b^{3} x^{4} - 4 \, {\left (a + \frac {b}{x^{2}}\right )} b^{4} x^{2} + b^{5}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {\frac {15 \, a^{5} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {15 \, {\left (a x^{2} + b\right )}^{\frac {7}{2}} a^{5} \mathrm {sgn}\left (x\right ) + 73 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} a^{5} b \mathrm {sgn}\left (x\right ) - 55 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{5} b^{2} \mathrm {sgn}\left (x\right ) + 15 \, \sqrt {a x^{2} + b} a^{5} b^{3} \mathrm {sgn}\left (x\right )}{a^{4} b x^{8}}}{384 \, a} \]
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Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^{5/2}}{x^4} \,d x \]
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