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

Optimal. Leaf size=19 \[ 2 a \sqrt{x}+\frac{2}{3} b x^{3/2} \]

[Out]

2*a*Sqrt[x] + (2*b*x^(3/2))/3

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Rubi [A]  time = 0.0036023, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ 2 a \sqrt{x}+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(3/2))/3

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0047361, size = 16, normalized size = 0.84 \[ \frac{2}{3} \sqrt{x} (3 a+b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*a + b*x))/3

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Maple [A]  time = 0.002, size = 13, normalized size = 0.7 \begin{align*}{\frac{2\,bx+6\,a}{3}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989671, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)

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Fricas [A]  time = 1.50122, size = 34, normalized size = 1.79 \begin{align*} \frac{2}{3} \,{\left (b x + 3 \, a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x + 3*a)*sqrt(x)

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Sympy [A]  time = 0.17349, size = 17, normalized size = 0.89 \begin{align*} 2 a \sqrt{x} + \frac{2 b x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*x**(3/2)/3

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Giac [A]  time = 1.2328, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)