Optimal. Leaf size=41 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a \sqrt {a+\frac {b}{x^2}}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{a b}\\ &=-\frac {1}{a \sqrt {a+\frac {b}{x^2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 1.51 \[ \frac {\sqrt {b} \sqrt {\frac {a x^2}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )-\sqrt {a} x}{a^{3/2} x \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 163, normalized size = 3.98 \[ \left [-\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - {\left (a x^{2} + b\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, {\left (a^{3} x^{2} + a^{2} b\right )}}, -\frac {a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} + {\left (a x^{2} + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a^{3} x^{2} + a^{2} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 1.54 \[ -\frac {\left (a \,x^{2}+b \right ) \left (a^{\frac {3}{2}} x -\sqrt {a \,x^{2}+b}\, a \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )\right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} a^{\frac {5}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 52, normalized size = 1.27 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {1}{\sqrt {a + \frac {b}{x^{2}}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 33, normalized size = 0.80 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a\,\sqrt {a+\frac {b}{x^2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.18, size = 187, normalized size = 4.56 \[ - \frac {2 a^{3} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{3} x^{2} \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{2} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{2}} \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2 a^{\frac {9}{2}} x^{2} + 2 a^{\frac {7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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