Optimal. Leaf size=26 \[ -\frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 26, 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 = {529, 23, 30}
\begin {gather*} -\frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 529
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x^2}}}{x \sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {\sqrt {-a+b x^2}}{x^2 \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^2} \, dx}{\sqrt {a-b x^2}}\\ &=-\frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 28, normalized size = 1.08
method | result | size |
gosper | \(-\frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{\sqrt {-b \,x^{2}+a}}\) | \(28\) |
default | \(-\frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{\sqrt {-b \,x^{2}+a}}\) | \(28\) |
risch | \(-\frac {i \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 7, normalized size = 0.27 \begin {gather*} \frac {i}{\sqrt {x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 41, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {-b x^{2} + a} {\left (x - 1\right )} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b x^{2} - a} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \frac {a}{x^{2}} + b}}{x \sqrt {a - b x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.34, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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