Optimal. Leaf size=114 \[ -\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{b x}-\frac {\sqrt {2} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2}} \]
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Rubi [F] time = 0.66, 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 \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}}}{x^2 \sqrt {b^2+a x^2}} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.24, size = 59, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {\sqrt {a x^2+b^2}+b} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )}{b x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 82, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{b x}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 38.63, size = 219, normalized size = 1.92 \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 \, b 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 \, b 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}}}}{\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 [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{x^{2} \sqrt {a \,x^{2}+b^{2}}}\, dx \end {gather*}
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}}}}{\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\,\sqrt {b^2+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.03, size = 44, normalized size = 0.39 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{\pi \sqrt {b} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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