Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}} \]
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Rubi [A]
time = 0.08, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2157, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}-b x^2}}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2157
Rubi steps
\begin {align*} \int \frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx &=\text {Subst}\left (\int \frac {1}{1+2 b x^2} \, dx,x,\frac {x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 51, normalized size = 1.06 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {-b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {2} \sqrt {b} x}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-b \,x^{2}+\sqrt {b^{2} x^{4}+a}}}{\sqrt {b^{2} x^{4}+a}}\, dx\]
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 = 1.25, size = 146, normalized size = 3.04 \begin {gather*} \left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{b}} \log \left (4 \, b^{2} x^{4} - 4 \, \sqrt {b^{2} x^{4} + a} b x^{2} + 2 \, {\left (\sqrt {2} b^{2} x^{3} \sqrt {-\frac {1}{b}} - \sqrt {2} \sqrt {b^{2} x^{4} + a} b x \sqrt {-\frac {1}{b}}\right )} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}} + a\right ), -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-b x^{2} + \sqrt {b^{2} x^{4} + a}}}{2 \, \sqrt {b} x}\right )}{2 \, \sqrt {b}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- b x^{2} + \sqrt {a + b^{2} x^{4}}}}{\sqrt {a + b^{2} x^{4}}}\, dx \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 = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {b^2\,x^4+a}-b\,x^2}}{\sqrt {b^2\,x^4+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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