Optimal. Leaf size=22 \[ \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}} \]
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Rubi [F]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\sqrt {a \,x^{2}+b^{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs.
\(2 (18) = 36\).
time = 0.40, size = 38, normalized size = 1.73 \begin {gather*} -\frac {2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}} {\left (b - \sqrt {a x^{2} + b^{2}}\right )}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (19) = 38\).
time = 0.41, size = 41, normalized size = 1.86 \begin {gather*} \frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\pi \sqrt {b} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{\sqrt {b^2+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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