Optimal. Leaf size=80 \[ \frac {1}{3} \left (2 x+\sqrt {2 x-1}\right )^{3/2}-\frac {1}{8} \left (2 \sqrt {2 x-1}+1\right ) \sqrt {2 x+\sqrt {2 x-1}}-\frac {3}{16} \sinh ^{-1}\left (\frac {2 \sqrt {2 x-1}+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {640, 612, 619, 215} \begin {gather*} \frac {1}{3} \left (2 x+\sqrt {2 x-1}\right )^{3/2}-\frac {1}{8} \left (2 \sqrt {2 x-1}+1\right ) \sqrt {2 x+\sqrt {2 x-1}}-\frac {3}{16} \sinh ^{-1}\left (\frac {2 \sqrt {2 x-1}+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \sqrt {2 x+\sqrt {-1+2 x}} \, dx &=\operatorname {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\frac {1}{3} \left (2 x+\sqrt {-1+2 x}\right )^{3/2}-\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\frac {1}{3} \left (2 x+\sqrt {-1+2 x}\right )^{3/2}-\frac {1}{8} \sqrt {2 x+\sqrt {-1+2 x}} \left (1+2 \sqrt {-1+2 x}\right )-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\frac {1}{3} \left (2 x+\sqrt {-1+2 x}\right )^{3/2}-\frac {1}{8} \sqrt {2 x+\sqrt {-1+2 x}} \left (1+2 \sqrt {-1+2 x}\right )-\frac {1}{16} \sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 \sqrt {-1+2 x}\right )\\ &=\frac {1}{3} \left (2 x+\sqrt {-1+2 x}\right )^{3/2}-\frac {1}{8} \sqrt {2 x+\sqrt {-1+2 x}} \left (1+2 \sqrt {-1+2 x}\right )-\frac {3}{16} \sinh ^{-1}\left (\frac {1+2 \sqrt {-1+2 x}}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.78 \begin {gather*} \frac {1}{48} \left (2 \sqrt {2 x+\sqrt {2 x-1}} \left (16 x+2 \sqrt {2 x-1}-3\right )-9 \sinh ^{-1}\left (\frac {2 \sqrt {2 x-1}+1}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 79, normalized size = 0.99 \begin {gather*} \frac {1}{24} \sqrt {2 x+\sqrt {2 x-1}} \left (8 (2 x-1)+2 \sqrt {2 x-1}+5\right )+\frac {3}{16} \log \left (-2 \sqrt {2 x-1}+2 \sqrt {2 x+\sqrt {2 x-1}}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 73, normalized size = 0.91 \begin {gather*} \frac {1}{24} \, {\left (16 \, x + 2 \, \sqrt {2 \, x - 1} - 3\right )} \sqrt {2 \, x + \sqrt {2 \, x - 1}} + \frac {3}{32} \, \log \left (-4 \, \sqrt {2 \, x + \sqrt {2 \, x - 1}} {\left (2 \, \sqrt {2 \, x - 1} + 1\right )} + 16 \, x + 8 \, \sqrt {2 \, x - 1} - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 67, normalized size = 0.84 \begin {gather*} \frac {1}{24} \, {\left (2 \, \sqrt {2 \, x - 1} {\left (4 \, \sqrt {2 \, x - 1} + 1\right )} + 5\right )} \sqrt {2 \, x + \sqrt {2 \, x - 1}} + \frac {3}{16} \, \log \left (2 \, \sqrt {2 \, x + \sqrt {2 \, x - 1}} - 2 \, \sqrt {2 \, x - 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.75 \begin {gather*} -\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {2 x -1}+\frac {1}{2}\right )}{3}\right )}{16}+\frac {\left (2 x +\sqrt {2 x -1}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {2 x -1}\right ) \sqrt {2 x +\sqrt {2 x -1}}}{8} \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 \sqrt {2 \, x + \sqrt {2 \, x - 1}}\,{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 \sqrt {2\,x+\sqrt {2\,x-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {2 x + \sqrt {2 x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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