3.9.60 \(\int \sqrt {1+x} \, dx\) [860]

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

[Out]

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

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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} \begin {gather*} \frac {2}{3} (x+1)^{3/2} \end {gather*}

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*}

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2))/3

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Maple [A]
time = 0.55, size = 8, normalized size = 0.73

method	result	size
gosper	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(8\)
derivativedivides	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(8\)
default	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(8\)
risch	\(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(8\)
trager	\(\left (\frac {2}{3}+\frac {2 x}{3}\right ) \sqrt {1+x}\)	\(12\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x +2\right ) \sqrt {1+x}}{3}}{2 \sqrt {\pi }}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 7, normalized size = 0.64 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

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 = 0.35, size = 7, normalized size = 0.64 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

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 = 0.01, size = 8, normalized size = 0.73 \begin {gather*} \frac {2 \left (x + 1\right )^{\frac {3}{2}}}{3} \end {gather*}

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 = 3.19, size = 7, normalized size = 0.64 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2),x)

[Out]

(2*(x + 1)^(3/2))/3

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