Optimal. Leaf size=26 \[ \frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b} \]
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Rubi [A]
time = 0.06, 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 = {2147, 30}
\begin {gather*} \frac {2 \sqrt {\sqrt {a+b^2 x^2}+b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2147
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx &=\frac {\text {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 = 0.04, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 2.13, size = 33, normalized size = 1.27 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b x+\sqrt {a+b^2 x^2}}}{b},b\text {!=}0\right \}\right \},\frac {x}{a^{\frac {1}{4}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b x +\sqrt {x^{2} b^{2}+a}}}{\sqrt {x^{2} b^{2}+a}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 22, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {b x + \sqrt {b^{2} x^{2} + a}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 27, normalized size = 1.04 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {b x+\sqrt {a+b^{2} x^{2}}}}{b} \end {gather*}
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 \begin {gather*} \frac {2\,\sqrt {\sqrt {b^2\,x^2+a}+b\,x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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