Optimal. Leaf size=28 \[ \frac {x^2 \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {515, 23, 8} \[ \frac {x^2 \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 23
Rule 515
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}}{\sqrt {a-b x^2}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int 1 \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\sqrt {b-\frac {a}{x^2}} x^2}{\sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {x^2 \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 41, normalized size = 1.46 \[ -\frac {\sqrt {-b x^{2} + a} x^{2} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b x^{2} - a} \]
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 \[ \int \frac {\sqrt {b - \frac {a}{x^{2}}} x}{\sqrt {-b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 1.50 \[ -\frac {\sqrt {\frac {b \,x^{2}-a}{x^{2}}}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{b \,x^{2}-a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.12, size = 7, normalized size = 0.25 \[ -i \, \sqrt {x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 27, normalized size = 0.96 \[ \frac {\sqrt {b\,x^2-a}\,\sqrt {x^2}}{\sqrt {a-b\,x^2}} \]
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 \[ \int \frac {x \sqrt {- \frac {a}{x^{2}} + b}}{\sqrt {a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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