\(\int (1+\sqrt {x}) \sqrt {x} \, dx\) [253]

Optimal result

Integrand size = 13, antiderivative size = 17 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {2 x^{3/2}}{3}+\frac {x^2}{2} \]

[Out]

2/3*x^(3/2)+1/2*x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {2 x^{3/2}}{3}+\frac {x^2}{2} \]

[In]

Int[(1 + Sqrt[x])*Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + x^2/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {x}+x\right ) \, dx \\ & = \frac {2 x^{3/2}}{3}+\frac {x^2}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {2 x^{3/2}}{3}+\frac {x^2}{2} \]

[In]

Integrate[(1 + Sqrt[x])*Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + x^2/2

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {x^{2}}{2}\)	\(12\)
default	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {x^{2}}{2}\)	\(12\)
trager	\(\frac {\left (-1+x \right ) \left (1+x \right )}{2}+\frac {2 x^{\frac {3}{2}}}{3}\)	\(15\)

[In]

int(x^(1/2)*(x^(1/2)+1),x,method=_RETURNVERBOSE)

[Out]

2/3*x^(3/2)+1/2*x^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {1}{2} \, x^{2} + \frac {2}{3} \, x^{\frac {3}{2}} \]

[In]

integrate(x^(1/2)*(1+x^(1/2)),x, algorithm="fricas")

[Out]

1/2*x^2 + 2/3*x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {2 x^{\frac {3}{2}}}{3} + \frac {x^{2}}{2} \]

[In]

integrate(x**(1/2)*(1+x**(1/2)),x)

[Out]

2*x**(3/2)/3 + x**2/2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {1}{2} \, {\left (\sqrt {x} + 1\right )}^{4} - \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{3} + {\left (\sqrt {x} + 1\right )}^{2} \]

[In]

integrate(x^(1/2)*(1+x^(1/2)),x, algorithm="maxima")

[Out]

1/2*(sqrt(x) + 1)^4 - 4/3*(sqrt(x) + 1)^3 + (sqrt(x) + 1)^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {1}{2} \, x^{2} + \frac {2}{3} \, x^{\frac {3}{2}} \]

[In]

integrate(x^(1/2)*(1+x^(1/2)),x, algorithm="giac")

[Out]

1/2*x^2 + 2/3*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \left (1+\sqrt {x}\right ) \sqrt {x} \, dx=\frac {x^2}{2}+\frac {2\,x^{3/2}}{3} \]

[In]

int(x^(1/2)*(x^(1/2) + 1),x)

[Out]

x^2/2 + (2*x^(3/2))/3