Optimal. Leaf size=26 \[ \frac {2 \sqrt {\sqrt {a+b^2 x^2}+b x}}{b} \]
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Rubi [A] time = 0.10, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2122, 30} \[ \frac {2 \sqrt {\sqrt {a+b^2 x^2}+b x}}{b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2122
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,b x+\sqrt {a+b^2 x^2}\right )}{b}\\ &=\frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ \frac {2 \sqrt {\sqrt {a+b^2 x^2}+b x}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 22, normalized size = 0.85 \[ \frac {2 \, \sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 22, normalized size = 0.85 \[ \frac {2 \, \sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x +\sqrt {b^{2} x^{2}+a}}}{\sqrt {b^{2} x^{2}+a}}\, dx \]
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 \[ \int \frac {\sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{\sqrt {b^{2} x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 22, normalized size = 0.85 \[ \frac {2\,\sqrt {\sqrt {b^2\,x^2+a}+b\,x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 27, normalized size = 1.04 \[ \begin {cases} \frac {2 \sqrt {b x + \sqrt {a + b^{2} x^{2}}}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt [4]{a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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