Optimal. Leaf size=110 \[ \frac {\sqrt {2} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{x} \]
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Rubi [F] time = 0.16, 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}}}{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}}}{x^2} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.26, size = 99, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (\sqrt {2} \sqrt {\sqrt {a x^2+b^2}-b} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-2 \sqrt {b}\right )}{2 \sqrt {b} x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 78, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 39.45, size = 213, normalized size = 1.94 \begin {gather*} \left [\frac {\sqrt {2} x \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x - 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {2} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 4 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{4 \, x}, -\frac {\sqrt {2} x \sqrt {\frac {a}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) + 2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{2 \, x}\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 {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 31, normalized size = 0.28
method | result | size |
meijerg | \(-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{x}\) | \(31\) |
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 {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{2}}\,{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 {\sqrt {b+\sqrt {b^2+a\,x^2}}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.91, size = 48, normalized size = 0.44 \begin {gather*} \frac {\sqrt {b} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{4 \pi x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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