3.2122 \(\int \left (a+b \sqrt{x}\right )^2 \, dx\)

Optimal. Leaf size=27 \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]

[Out]

a^2*x + (4*a*b*x^(3/2))/3 + (b^2*x^2)/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0438566, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2,x]

[Out]

a^2*x + (4*a*b*x^(3/2))/3 + (b^2*x^2)/2

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a b x^{\frac{3}{2}}}{3} + b^{2} \int x\, dx + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*b*x**(3/2)/3 + b**2*Integral(x, x) + Integral(a**2, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00780663, size = 27, normalized size = 1. \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^2,x]

[Out]

a^2*x + (4*a*b*x^(3/2))/3 + (b^2*x^2)/2

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 22, normalized size = 0.8 \[ x{a}^{2}+{\frac{4\,ab}{3}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^2,x)

[Out]

x*a^2+4/3*a*b*x^(3/2)+1/2*b^2*x^2

_______________________________________________________________________________________

Maxima [A]  time = 1.43792, size = 28, normalized size = 1.04 \[ \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*x^2 + 4/3*a*b*x^(3/2) + a^2*x

_______________________________________________________________________________________

Fricas [A]  time = 0.230621, size = 28, normalized size = 1.04 \[ \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*x^2 + 4/3*a*b*x^(3/2) + a^2*x

_______________________________________________________________________________________

Sympy [A]  time = 0.353698, size = 24, normalized size = 0.89 \[ a^{2} x + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{b^{2} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**2,x)

[Out]

a**2*x + 4*a*b*x**(3/2)/3 + b**2*x**2/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216161, size = 28, normalized size = 1.04 \[ \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*x^2 + 4/3*a*b*x^(3/2) + a^2*x