Integrand size = 15, antiderivative size = 95 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{16 b x}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{16 b^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 285, 327, 223, 212} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{16 b^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{16 b x}-\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3} \]
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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 )^{3/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {1}{2} a \text {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {1}{8} a^2 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{16 b x}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )}{16 b} \\ & = -\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{16 b x}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )}{16 b} \\ & = -\frac {a \sqrt {a+\frac {b}{x^2}}}{8 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{16 b x}+\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{16 b^{3/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (8 b^2+14 a b x^2+3 a^2 x^4\right )+\frac {3 a^3 x^6 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{48 b^{3/2} x^5} \]
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Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\left (3 a^{2} x^{4}+14 a b \,x^{2}+8 b^{2}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{48 x^{5} b}+\frac {a^{3} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{16 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\) | \(98\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{3} x^{6}-\left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{3} x^{6}+\left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{2} x^{4}-3 \sqrt {a \,x^{2}+b}\, a^{3} b \,x^{6}+2 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a b \,x^{2}-8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{2}\right )}{48 x^{3} \left (a \,x^{2}+b \right )^{\frac {3}{2}} b^{3}}\) | \(145\) |
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Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=\left [\frac {3 \, a^{3} \sqrt {b} x^{5} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{96 \, b^{2} x^{5}}, -\frac {3 \, a^{3} \sqrt {-b} x^{5} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{48 \, b^{2} x^{5}}\right ] \]
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Time = 3.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=- \frac {a^{\frac {5}{2}}}{16 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {17 a^{\frac {3}{2}}}{48 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {11 \sqrt {a} b}{24 x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{16 b^{\frac {3}{2}}} - \frac {b^{2}}{6 \sqrt {a} x^{7} \sqrt {1 + \frac {b}{a x^{2}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (75) = 150\).
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=-\frac {a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{32 \, b^{\frac {3}{2}}} - \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{3} x^{5} + 8 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{3} b x^{3} - 3 \, \sqrt {a + \frac {b}{x^{2}}} a^{3} b^{2} x}{48 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{3} b x^{6} - 3 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b^{2} x^{4} + 3 \, {\left (a + \frac {b}{x^{2}}\right )} b^{3} x^{2} - b^{4}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=-\frac {\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {3 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} a^{4} \mathrm {sgn}\left (x\right ) + 8 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{4} b \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {a x^{2} + b} a^{4} b^{2} \mathrm {sgn}\left (x\right )}{a^{3} b x^{6}}}{48 \, a} \]
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Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^{3/2}}{x^4} \,d x \]
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