Optimal. Leaf size=29 \[ \frac {4}{3} \left (\sqrt {x}+1\right )^{3/2}-4 \sqrt {\sqrt {x}+1} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {4}{3} \left (\sqrt {x}+1\right )^{3/2}-4 \sqrt {\sqrt {x}+1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-4 \sqrt {1+\sqrt {x}}+\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.76 \[ \frac {4}{3} \left (\sqrt {x}-2\right ) \sqrt {\sqrt {x}+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 14, normalized size = 0.48 \[ \frac {4}{3} \, \sqrt {\sqrt {x} + 1} {\left (\sqrt {x} - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 19, normalized size = 0.66 \[ \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.69 \[ \frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-4 \sqrt {\sqrt {x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 19, normalized size = 0.66 \[ \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 12, normalized size = 0.41 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.91, size = 117, normalized size = 4.03 \[ - \frac {4 x^{\frac {5}{2}} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{\frac {5}{2}}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 x^{3} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{2} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{3 x^{\frac {5}{2}} + 3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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