3.1649 \(\int (a+\frac{b}{x}) \sqrt{x} \, dx\)

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

[Out]

2*b*Sqrt[x] + (2*a*x^(3/2))/3

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Rubi [A]  time = 0.0043763, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{3} a x^{3/2}+2 b \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*Sqrt[x],x]

[Out]

2*b*Sqrt[x] + (2*a*x^(3/2))/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0041841, size = 16, normalized size = 0.84 \[ \frac{2}{3} \sqrt{x} (a x+3 b) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*b + a*x))/3

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Maple [A]  time = 0.001, size = 13, normalized size = 0.7 \begin{align*}{\frac{2\,ax+6\,b}{3}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.974236, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \,{\left (a + \frac{3 \, b}{x}\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a + 3*b/x)*x^(3/2)

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Fricas [A]  time = 1.92147, size = 34, normalized size = 1.79 \begin{align*} \frac{2}{3} \,{\left (a x + 3 \, b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a*x + 3*b)*sqrt(x)

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Sympy [A]  time = 0.194288, size = 17, normalized size = 0.89 \begin{align*} \frac{2 a x^{\frac{3}{2}}}{3} + 2 b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(3/2)/3 + 2*b*sqrt(x)

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Giac [A]  time = 1.09587, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, a x^{\frac{3}{2}} + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a*x^(3/2) + 2*b*sqrt(x)