3.2115 \(\int (a+b \sqrt{x}) \, dx\)

Optimal. Leaf size=14 \[ a x+\frac{2}{3} b x^{3/2} \]

[Out]

a*x + (2*b*x^(3/2))/3

________________________________________________________________________________________

Rubi [A]  time = 0.0022963, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right ) \, dx &=a x+\frac{2}{3} b x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0014288, size = 14, normalized size = 1. \[ a x+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sqrt[x],x]

[Out]

a*x + (2*b*x^(3/2))/3

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*} ax+{\frac{2\,b}{3}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*x^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.96507, size = 14, normalized size = 1. \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="maxima")

[Out]

2/3*b*x^(3/2) + a*x

________________________________________________________________________________________

Fricas [A]  time = 1.4622, size = 28, normalized size = 2. \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="fricas")

[Out]

2/3*b*x^(3/2) + a*x

________________________________________________________________________________________

Sympy [A]  time = 0.052043, size = 12, normalized size = 0.86 \begin{align*} a x + \frac{2 b x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x**(1/2),x)

[Out]

a*x + 2*b*x**(3/2)/3

________________________________________________________________________________________

Giac [A]  time = 1.08628, size = 14, normalized size = 1. \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*x^(1/2),x, algorithm="giac")

[Out]

2/3*b*x^(3/2) + a*x