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

Optimal result

Integrand size = 9, antiderivative size = 13 \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} (1+2 x)^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} (2 x+1)^{3/2} \]

[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*} \text {integral}& = \frac {1}{3} (1+2 x)^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} (1+2 x)^{3/2} \]

[In]

Integrate[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
derivativedivides	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
default	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
risch	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
pseudoelliptic	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
trager	\(\left (\frac {1}{3}+\frac {2 x}{3}\right ) \sqrt {1+2 x}\)	\(14\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+4 x \right ) \sqrt {1+2 x}}{3}}{4 \sqrt {\pi }}\)	\(29\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \sqrt {1+2 x} \, dx=\frac {\left (2 x + 1\right )^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

(2*x + 1)**(3/2)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \sqrt {1+2 x} \, dx=\frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \sqrt {1+2 x} \, dx=\frac {{\left (2\,x+1\right )}^{3/2}}{3} \]

[In]

int((2*x + 1)^(1/2),x)

[Out]

(2*x + 1)^(3/2)/3