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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{3} (2 x+1)^{3/2} \end {gather*}

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*}

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Mathematica [A]
time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{3} (1+2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

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Mathics [A]
time = 1.59, size = 9, normalized size = 0.69 \begin {gather*} \frac {\left (1+2 x\right )^{\frac {3}{2}}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sqrt[2*x + 1],x]')

[Out]

(1 + 2 x) ^ (3 / 2) / 3

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Maple [A]
time = 0.10, size = 10, normalized size = 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	\(\frac {\left (1+2 x \right )^{\frac {3}{2}}}{3}\)	\(10\)
trager	\(\left (\frac {1}{3}+\frac {2 x}{3}\right ) \sqrt {1+2 x}\)	\(14\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+4 x \right ) \sqrt {1+2 x}}{3}}{4 \sqrt {\pi }}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

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Fricas [A]
time = 0.32, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

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Sympy [A]
time = 0.03, size = 8, normalized size = 0.62 \begin {gather*} \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{3} \end {gather*}

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

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Giac [A]
time = 0.00, size = 17, normalized size = 1.31 \begin {gather*} \frac {1}{3} \sqrt {2 x+1} \left (2 x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x + 1)^(3/2)

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Mupad [B]
time = 0.32, size = 9, normalized size = 0.69 \begin {gather*} \frac {{\left (2\,x+1\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(1/2),x)

[Out]

(2*x + 1)^(3/2)/3

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