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

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 = 0.00429097, 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 \[ \frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2))/3

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Rubi in Sympy [A]  time = 0.51193, size = 8, normalized size = 0.73 \[ \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/2),x)

[Out]

2*(x + 1)**(3/2)/3

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Mathematica [A]  time = 0.002221, 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 = 0.003, size = 8, normalized size = 0.7 \[{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

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 = 0.678469, size = 9, normalized size = 0.82 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1),x, algorithm="maxima")

[Out]

2/3*(x + 1)^(3/2)

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Fricas [A]  time = 0.267706, size = 9, normalized size = 0.82 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1),x, algorithm="fricas")

[Out]

2/3*(x + 1)^(3/2)

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Sympy [A]  time = 0.06315, size = 8, normalized size = 0.73 \[ \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} \]

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/XCAS [A]  time = 0.261984, size = 9, normalized size = 0.82 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1),x, algorithm="giac")

[Out]

2/3*(x + 1)^(3/2)