3.860 \(\int \sqrt {1+x} \, dx\)

Optimal. Leaf size=11 \[ \frac {2}{3} (x+1)^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \frac {2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2))/3

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \sqrt {1+x} \, dx &=\frac {2}{3} (1+x)^{3/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2))/3

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 7, normalized size = 0.64 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2)

________________________________________________________________________________________

giac [A]  time = 0.32, size = 7, normalized size = 0.64 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 8, normalized size = 0.73 \[ \frac {2 \left (x +1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.88, size = 7, normalized size = 0.64 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 3.42, size = 7, normalized size = 0.64 \[ \frac {2\,{\left (x+1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2),x)

[Out]

(2*(x + 1)^(3/2))/3

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 8, normalized size = 0.73 \[ \frac {2 \left (x + 1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x + 1)**(3/2)/3

________________________________________________________________________________________