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 = 0.0133593, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 247
Rule 190
Rule 43
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*}
Mathematica [A] time = 0.0103312, 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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Maple [A] time = 0.004, size = 28, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( 3+\sqrt{2\,x-1} \right ) ^{{\frac{3}{2}}}}-6\,\sqrt{3+\sqrt{2\,x-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07937, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43197, size = 66, normalized size = 1.78 \begin{align*} \frac{2}{3} \, \sqrt{\sqrt{2 \, x - 1} + 3}{\left (\sqrt{2 \, x - 1} - 6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.960737, size = 265, normalized size = 7.16 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20493, size = 43, normalized size = 1.16 \begin{align*} \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} + 4 \, \sqrt{3} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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