3.1 \(\int \sqrt{1+2 x} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{3} (2 x+1)^{3/2} \]

[Out]

(1 + 2*x)^(3/2)/3

________________________________________________________________________________________

Rubi [A]  time = 0.0009888, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{1}{3} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + 2*x],x]

[Out]

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

Mathematica [A]  time = 0.0371224, size = 13, normalized size = 1. \[ \frac{1}{3} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{3} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.944664, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 0.416858, size = 28, normalized size = 2.15 \begin{align*} \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 0.051242, size = 8, normalized size = 0.62 \begin{align*} \frac{\left (2 x + 1\right )^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x + 1)**(3/2)/3

________________________________________________________________________________________

Giac [A]  time = 1.06262, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)