Optimal. Leaf size=144 \[ -\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{4 b x}-\frac {1}{2 x \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{2 \sqrt {2} b^{3/2}} \]
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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 {1}{x^2 \sqrt {b+\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 {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx &=\int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.40, size = 143, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (9 a x^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )+2 b \left (\sqrt {a x^2+b^2}+b\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )+2 b \left (5 \sqrt {a x^2+b^2}-7 b\right )\right )}{24 a b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 112, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{4 b x}-\frac {1}{2 x \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{4 \sqrt {2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 44.20, size = 282, normalized size = 1.96 \begin {gather*} \left [\frac {\sqrt {\frac {1}{2}} a x^{3} \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 + 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {\frac {1}{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 ) - 2 \, {\left (a x^{2} - 2 \, b^{2} + 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{8 \, a b x^{3}}, \frac {\sqrt {\frac {1}{2}} a x^{3} \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - {\left (a x^{2} - 2 \, b^{2} + 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{4 \, a b x^{3}}\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 {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 = 0.22 \begin {gather*} -\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}, \frac {3}{4}\right ], \left [\frac {1}{2}, \frac {3}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{2 \left (b^{2}\right )^{\frac {1}{4}} x} \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 {1}{\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 {1}{x^2\,\sqrt {b+\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 = 0.99, size = 46, normalized size = 0.32 \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 {3}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{2 \pi \sqrt {b} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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