Optimal. Leaf size=34 \[ -\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \]
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Rubi [A] time = 0.01, antiderivative size = 34, 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, 325, 205} \[ -\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^4} \, dx &=\int \frac {1}{x^2 \left (b+a x^2\right )} \, dx\\ &=-\frac {1}{b x}-\frac {a \int \frac {1}{b+a x^2} \, dx}{b}\\ &=-\frac {1}{b x}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ -\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 82, normalized size = 2.41 \[ \left [\frac {x \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) - 2}{2 \, b x}, -\frac {x \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) + 1}{b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.85 \[ -\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.88 \[ -\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.84, size = 29, normalized size = 0.85 \[ -\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.76 \[ -\frac {1}{b\,x}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 65, normalized size = 1.91 \[ \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (x - \frac {b^{2} \sqrt {- \frac {a}{b^{3}}}}{a} \right )}}{2} - \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (x + \frac {b^{2} \sqrt {- \frac {a}{b^{3}}}}{a} \right )}}{2} - \frac {1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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