3.1789 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} \sqrt{x}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0134833, antiderivative size = 44, 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} \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*Sqrt[x]),x]

[Out]

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

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

Mathematica [A]  time = 0.0131979, size = 28, normalized size = 0.64 \[ \frac{2 (a x+2 b)}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*Sqrt[x]),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.985089, size = 49, normalized size = 1.11 \begin{align*} \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a^{2}} + \frac{2 \, b}{\sqrt{a + \frac{b}{x}} a^{2} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4674, size = 77, normalized size = 1.75 \begin{align*} \frac{2 \,{\left (a x + 2 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{a^{3} x + a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.8317, size = 39, normalized size = 0.89 \begin{align*} \frac{2 x}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a x}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16399, size = 42, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (\sqrt{a x + b} + \frac{b}{\sqrt{a x + b}}\right )}}{a^{2}} - \frac{4 \, \sqrt{b}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(1/2),x, algorithm="giac")

[Out]

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