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 = 0.00507333, 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 \[ \frac{1}{3} (2 x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + 2*x],x]

[Out]

(1 + 2*x)^(3/2)/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.517797, size = 8, normalized size = 0.62 \[ \frac{\left (2 x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(1/2),x)

[Out]

(2*x + 1)**(3/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.00289457, 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 = 0.264, size = 10, normalized size = 0.8 \[{\frac{1}{3} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}} \]

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 = 1.3247, size = 12, normalized size = 0.92 \[ \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x + 1),x, algorithm="maxima")

[Out]

1/3*(2*x + 1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.208579, size = 12, normalized size = 0.92 \[ \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x + 1),x, algorithm="fricas")

[Out]

1/3*(2*x + 1)^(3/2)

_______________________________________________________________________________________

Sympy [A]  time = 0.031499, size = 8, normalized size = 0.62 \[ \frac{\left (2 x + 1\right )^{\frac{3}{2}}}{3} \]

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/XCAS [A]  time = 0.231308, size = 12, normalized size = 0.92 \[ \frac{1}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x + 1),x, algorithm="giac")

[Out]

1/3*(2*x + 1)^(3/2)