Optimal. Leaf size=62 \[ \frac {2}{3} \left (x+\sqrt {x}+1\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {x+\sqrt {x}+1}-\frac {3}{8} \sinh ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1341, 640, 612, 619, 215} \[ \frac {2}{3} \left (x+\sqrt {x}+1\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {x+\sqrt {x}+1}-\frac {3}{8} \sinh ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 612
Rule 619
Rule 640
Rule 1341
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {x}+x} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\operatorname {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {1}{8} \sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 \sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {3}{8} \sinh ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 0.79 \[ \frac {1}{24} \left (2 \sqrt {x+\sqrt {x}+1} \left (8 x+2 \sqrt {x}+5\right )-9 \sinh ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 51, normalized size = 0.82 \[ \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} + 5\right )} \sqrt {x + \sqrt {x} + 1} + \frac {3}{16} \, \log \left (4 \, \sqrt {x + \sqrt {x} + 1} {\left (2 \, \sqrt {x} + 1\right )} - 8 \, x - 8 \, \sqrt {x} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 45, normalized size = 0.73 \[ \frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} + 5\right )} \sqrt {x + \sqrt {x} + 1} + \frac {3}{8} \, \log \left (2 \, \sqrt {x + \sqrt {x} + 1} - 2 \, \sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 42, normalized size = 0.68 \[ -\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {x}+\frac {1}{2}\right )}{3}\right )}{8}+\frac {2 \left (x +\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {x}+1\right ) \sqrt {x +\sqrt {x}+1}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x + \sqrt {x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {x+\sqrt {x}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x} + x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________