Optimal. Leaf size=37 \[ \frac {2}{3} \left (\sqrt {2 x-1}+3\right )^{3/2}-6 \sqrt {\sqrt {2 x-1}+3} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {247, 190, 43} \[ \frac {2}{3} \left (\sqrt {2 x-1}+3\right )^{3/2}-6 \sqrt {\sqrt {2 x-1}+3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+\sqrt {-1+2 x}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {x}}} \, dx,x,-1+2 x\right )\\ &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {3+x}} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {3}{\sqrt {3+x}}+\sqrt {3+x}\right ) \, dx,x,\sqrt {-1+2 x}\right )\\ &=-6 \sqrt {3+\sqrt {-1+2 x}}+\frac {2}{3} \left (3+\sqrt {-1+2 x}\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.81 \[ \frac {2}{3} \left (\sqrt {2 x-1}-6\right ) \sqrt {\sqrt {2 x-1}+3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 22, normalized size = 0.59 \[ \frac {2}{3} \, \sqrt {\sqrt {2 \, x - 1} + 3} {\left (\sqrt {2 \, x - 1} - 6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 27, normalized size = 0.73 \[ \frac {2}{3} \, {\left (\sqrt {2 \, x - 1} + 3\right )}^{\frac {3}{2}} - 6 \, \sqrt {\sqrt {2 \, x - 1} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.76 \[ \frac {2 \left (3+\sqrt {2 x -1}\right )^{\frac {3}{2}}}{3}-6 \sqrt {3+\sqrt {2 x -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 27, normalized size = 0.73 \[ \frac {2}{3} \, {\left (\sqrt {2 \, x - 1} + 3\right )}^{\frac {3}{2}} - 6 \, \sqrt {\sqrt {2 \, x - 1} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 24, normalized size = 0.65 \[ \frac {\sqrt {3}\,\left (2\,x-1\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {\sqrt {2\,x-1}}{3}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.16, size = 265, normalized size = 7.16 \[ - \frac {6 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} \sqrt {\sqrt {2} \sqrt {x - \frac {1}{2}} + 3}}{3 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 9 \sqrt {3} \left (x - \frac {1}{2}\right )^{2}} + \frac {36 \sqrt {2} \left (x - \frac {1}{2}\right )^{\frac {5}{2}}}{3 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 9 \sqrt {3} \left (x - \frac {1}{2}\right )^{2}} + \frac {4 \sqrt {3} \left (x - \frac {1}{2}\right )^{3} \sqrt {\sqrt {2} \sqrt {x - \frac {1}{2}} + 3}}{3 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 9 \sqrt {3} \left (x - \frac {1}{2}\right )^{2}} - \frac {36 \sqrt {3} \left (x - \frac {1}{2}\right )^{2} \sqrt {\sqrt {2} \sqrt {x - \frac {1}{2}} + 3}}{3 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 9 \sqrt {3} \left (x - \frac {1}{2}\right )^{2}} + \frac {108 \left (x - \frac {1}{2}\right )^{2}}{3 \sqrt {6} \left (x - \frac {1}{2}\right )^{\frac {5}{2}} + 9 \sqrt {3} \left (x - \frac {1}{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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