Optimal. Leaf size=19 \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]
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Rubi [A] time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2122, 30} \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2122
Rubi steps
\begin {align*} \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,x+\sqrt {a^2+x^2}\right )\\ &=2 \sqrt {x+\sqrt {a^2+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 15, normalized size = 0.79 \[ 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.79, size = 15, normalized size = 0.79 \[ 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x +\sqrt {a^{2}+x^{2}}}}{\sqrt {a^{2}+x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a^{2} + x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 15, normalized size = 0.79 \[ 2\,\sqrt {x+\sqrt {a^2+x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 15, normalized size = 0.79 \[ 2 \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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