Optimal. Leaf size=22 \[ \frac {\sqrt {x^2+1} \sqrt {a x^2}}{x} \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 261} \[ \frac {\sqrt {x^2+1} \sqrt {a x^2}}{x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 261
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2}}{\sqrt {1+x^2}} \, dx &=\frac {\sqrt {a x^2} \int \frac {x}{\sqrt {1+x^2}} \, dx}{x}\\ &=\frac {\sqrt {a x^2} \sqrt {1+x^2}}{x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ \frac {\sqrt {x^2+1} \sqrt {a x^2}}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 0.82 \[ \frac {\sqrt {a x^{2}} \sqrt {x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 19, normalized size = 0.86 \[ {\left (\sqrt {x^{2} + 1} \mathrm {sgn}\relax (x) - \mathrm {sgn}\relax (x)\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a \,x^{2}}\, \sqrt {x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a} x^{2} + \sqrt {a}}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a}\,\sqrt {x^2+1}\,\sqrt {x^2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 20, normalized size = 0.91 \[ \frac {\sqrt {a} \sqrt {x^{2} + 1} \sqrt {x^{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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