Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {a \sqrt {a+\frac {b}{x^2}}}{b^2}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {a \sqrt {a+\frac {b}{x^2}}}{b^2}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {a \sqrt {a+\frac {b}{x^2}}}{b^2}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-b+2 a x^2\right )}{3 b^2 x^2} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03
method | result | size |
trager | \(\frac {\left (2 a \,x^{2}-b \right ) \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 x^{2} b^{2}}\) | \(36\) |
gosper | \(\frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}-b \right )}{3 x^{4} b^{2} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(39\) |
default | \(\frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}-b \right )}{3 x^{4} b^{2} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(39\) |
risch | \(\frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}-b \right )}{3 x^{4} b^{2} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(39\) |
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {{\left (2 \, a x^{2} - b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, b^{2} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (29) = 58\).
Time = 0.77 (sec) , antiderivative size = 231, normalized size of antiderivative = 6.60 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {2 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{5} + 3 a^{\frac {3}{2}} b^{4} x^{3}} + \frac {a^{\frac {5}{2}} b^{\frac {5}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{5} + 3 a^{\frac {3}{2}} b^{4} x^{3}} - \frac {a^{\frac {3}{2}} b^{\frac {7}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{5} + 3 a^{\frac {3}{2}} b^{4} x^{3}} - \frac {2 a^{4} b x^{5}}{3 a^{\frac {5}{2}} b^{3} x^{5} + 3 a^{\frac {3}{2}} b^{4} x^{3}} - \frac {2 a^{3} b^{2} x^{3}}{3 a^{\frac {5}{2}} b^{3} x^{5} + 3 a^{\frac {3}{2}} b^{4} x^{3}} \]
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Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=-\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}}}{3 \, b^{2}} + \frac {\sqrt {a + \frac {b}{x^{2}}} a}{b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=\frac {4 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )} a^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{3} \mathrm {sgn}\left (x\right )} \]
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Time = 6.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^5} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}\,\left (b-2\,a\,x^2\right )}{3\,b^2\,x^2} \]
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