Optimal. Leaf size=31 \[ -\frac {\sqrt {b-\frac {a}{x^2}}}{2 x \sqrt {a-b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ -\frac {\sqrt {b-\frac {a}{x^2}}}{2 x \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 515
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x^2}}}{x^2 \sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {\sqrt {-a+b x^2}}{x^3 \sqrt {a-b x^2}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {1}{x^3} \, dx}{\sqrt {a-b x^2}}\\ &=-\frac {\sqrt {b-\frac {a}{x^2}}}{2 x \sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {\sqrt {b-\frac {a}{x^2}}}{2 x \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 44, normalized size = 1.42 \[ -\frac {\sqrt {-b x^{2} + a} {\left (x^{2} - 1\right )} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{2 \, {\left (b x^{3} - a x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.37, size = 5, normalized size = 0.16 \[ \frac {i}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.00 \[ -\frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{2 \sqrt {-b \,x^{2}+a}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.06, size = 5, normalized size = 0.16 \[ \frac {i}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 25, normalized size = 0.81 \[ -\frac {\sqrt {b-\frac {a}{x^2}}}{2\,x\,\sqrt {a-b\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {a}{x^{2}} + b}}{x^{2} \sqrt {a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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