3.1780 \(\int \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]

[Out]

(-4*b*Sqrt[a + b/x]*Sqrt[x])/(3*a^2) + (2*Sqrt[a + b/x]*x^(3/2))/(3*a)

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Rubi [A]  time = 0.0125094, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[a + b/x],x]

[Out]

(-4*b*Sqrt[a + b/x]*Sqrt[x])/(3*a^2) + (2*Sqrt[a + b/x]*x^(3/2))/(3*a)

Rule 271

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

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 \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx &=\frac{2 \sqrt{a+\frac{b}{x}} x^{3/2}}{3 a}-\frac{(2 b) \int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx}{3 a}\\ &=-\frac{4 b \sqrt{a+\frac{b}{x}} \sqrt{x}}{3 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{3/2}}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0168336, size = 30, normalized size = 0.62 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x-2 b)}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[a + b/x],x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-2*b + a*x))/(3*a^2)

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Maple [A]  time = 0.003, size = 32, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( ax-2\,b \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948021, size = 46, normalized size = 0.96 \begin{align*} \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 3 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42539, size = 63, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (a x - 2 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.47776, size = 42, normalized size = 0.88 \begin{align*} \frac{2 \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3 a} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1704, size = 43, normalized size = 0.9 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{3 \, a^{2}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x + b} b\right )}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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