3.1755 \(\int \sqrt{a+\frac{b}{x}} \sqrt{x} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{3 a} \]

[Out]

(2*(a + b/x)^(3/2)*x^(3/2))/(3*a)

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Rubi [A]  time = 0.0053108, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*Sqrt[x],x]

[Out]

(2*(a + b/x)^(3/2)*x^(3/2))/(3*a)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0102275, size = 23, normalized size = 1. \[ \frac{2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*Sqrt[x],x]

[Out]

(2*(a + b/x)^(3/2)*x^(3/2))/(3*a)

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Maple [A]  time = 0.003, size = 25, normalized size = 1.1 \begin{align*}{\frac{2\,ax+2\,b}{3\,a}\sqrt{{\frac{ax+b}{x}}}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988956, size = 23, normalized size = 1. \begin{align*} \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a + b/x)^(3/2)*x^(3/2)/a

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Fricas [A]  time = 1.4884, size = 58, normalized size = 2.52 \begin{align*} \frac{2 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.99468, size = 39, normalized size = 1.7 \begin{align*} \frac{2 \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3} + \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.1295, size = 32, normalized size = 1.39 \begin{align*} \frac{2}{3} \,{\left (\frac{{\left (a x + b\right )}^{\frac{3}{2}}}{a} - \frac{b^{\frac{3}{2}}}{a}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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