Optimal. Leaf size=98 \[ \frac {x}{2 \sqrt {\sqrt {a^2 x^4+b}+a x^2}}+\frac {\log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \]
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Rubi [F] time = 0.04, 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 {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx &=\int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.56, size = 98, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {\log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {2} \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.56, size = 229, normalized size = 2.34 \begin {gather*} \left [\frac {\frac {\sqrt {2} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{\sqrt {a}} - 4 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{8 \, b}, -\frac {\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-\frac {1}{a}}}{2 \, x}\right ) + 2 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{4 \, b}\right ] \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*} \int \frac {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}\, dx\]
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*} \int \frac {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}} \,d x \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 {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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