Optimal. Leaf size=30 \[ \frac {\sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ),-1\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {116} \[ \frac {\sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 116
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {2-b x} \sqrt {2+b x}} \, dx &=\frac {\sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.87 \[ \sqrt {x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b^2 x^2}{4}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {x}}{b^{2} x^{3} - 4 \, x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.07 \[ \frac {\sqrt {-b x}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {b x +2}}{2}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {x}} \]
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 \[ \int \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {x}\,\sqrt {2-b\,x}\,\sqrt {b\,x+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.65, size = 95, normalized size = 3.17 \[ \frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b}} - \frac {\sqrt {2} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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