Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {x}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 288, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {x}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 288
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^2} \, dx &=\int \frac {x^2}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac {x}{2 a \left (b+a x^2\right )}+\frac {\int \frac {1}{b+a x^2} \, dx}{2 a}\\ &=-\frac {x}{2 a \left (b+a x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {x}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 120, normalized size = 2.67 \[ \left [-\frac {2 \, a b x + {\left (a x^{2} + b\right )} \sqrt {-a b} \log \left (\frac {a x^{2} - 2 \, \sqrt {-a b} x - b}{a x^{2} + b}\right )}{4 \, {\left (a^{3} b x^{2} + a^{2} b^{2}\right )}}, -\frac {a b x - {\left (a x^{2} + b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{b}\right )}{2 \, {\left (a^{3} b x^{2} + a^{2} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.78 \[ \frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {x}{2 \, {\left (a x^{2} + b\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \[ -\frac {x}{2 \left (a \,x^{2}+b \right ) a}+\frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 36, normalized size = 0.80 \[ -\frac {x}{2 \, {\left (a^{2} x^{2} + a b\right )}} + \frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 33, normalized size = 0.73 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{2\,a^{3/2}\,\sqrt {b}}-\frac {x}{2\,a\,\left (a\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 78, normalized size = 1.73 \[ - \frac {x}{2 a^{2} x^{2} + 2 a b} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a b \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a b \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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