Optimal. Leaf size=46 \[ \frac {4}{7} \left (\sqrt {x}+1\right )^{7/2}-\frac {8}{5} \left (\sqrt {x}+1\right )^{5/2}+\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {4}{7} \left (\sqrt {x}+1\right )^{7/2}-\frac {8}{5} \left (\sqrt {x}+1\right )^{5/2}+\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx &=2 \operatorname {Subst}\left (\int x^2 \sqrt {1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\sqrt {1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {x}\right )^{7/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.59 \[ \frac {4}{105} \left (\sqrt {x}+1\right )^{3/2} \left (15 x-12 \sqrt {x}+8\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 23, normalized size = 0.50 \[ \frac {4}{105} \, {\left ({\left (15 \, x - 4\right )} \sqrt {x} + 3 \, x + 8\right )} \sqrt {\sqrt {x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 28, normalized size = 0.61 \[ \frac {4}{7} \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - \frac {8}{5} \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 29, normalized size = 0.63 \[ \frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-\frac {8 \left (\sqrt {x}+1\right )^{\frac {5}{2}}}{5}+\frac {4 \left (\sqrt {x}+1\right )^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.86, size = 28, normalized size = 0.61 \[ \frac {4}{7} \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - \frac {8}{5} \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {x}\,\sqrt {\sqrt {x}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.39, size = 398, normalized size = 8.65 \[ \frac {60 x^{\frac {15}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {200 x^{\frac {13}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {60 x^{\frac {11}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {96 x^{\frac {11}{2}}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {32 x^{\frac {9}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {32 x^{\frac {9}{2}}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {192 x^{7} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {80 x^{6} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {32 x^{6}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {80 x^{5} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {96 x^{5}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________