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

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

[Out]

(2*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.0006516, antiderivative size = 11, normalized size of antiderivative = 1., 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 = 0.0021514, size = 11, normalized size = 1. \[ \frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2))/3

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Maple [A]  time = 0.001, size = 8, normalized size = 0.7 \begin{align*}{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992441, size = 9, normalized size = 0.82 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

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)

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Fricas [A]  time = 1.41257, size = 26, normalized size = 2.36 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

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)

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Sympy [A]  time = 0.051079, size = 8, normalized size = 0.73 \begin{align*} \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x + 1)**(3/2)/3

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Giac [A]  time = 1.10494, size = 9, normalized size = 0.82 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

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)