3.1788 \(\int \frac{\sqrt{x}}{(a+\frac{b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(-16*b^2)/(3*a^3*Sqrt[a + b/x]*Sqrt[x]) - (8*b*Sqrt[x])/(3*a^2*Sqrt[a + b/x]) + (2*x^(3/2))/(3*a*Sqrt[a + b/x]
)

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Rubi [A]  time = 0.0227034, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(-16*b^2)/(3*a^3*Sqrt[a + b/x]*Sqrt[x]) - (8*b*Sqrt[x])/(3*a^2*Sqrt[a + b/x]) + (2*x^(3/2))/(3*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{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx &=\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}}-\frac{(4 b) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}} \, dx}{3 a}\\ &=-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}}+\frac{\left (8 b^2\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{3/2}} \, dx}{3 a^2}\\ &=-\frac{16 b^2}{3 a^3 \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0120752, size = 41, normalized size = 0.55 \[ \frac{2 \left (a^2 x^2-4 a b x-8 b^2\right )}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(2*(-8*b^2 - 4*a*b*x + a^2*x^2))/(3*a^3*Sqrt[a + b/x]*Sqrt[x])

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Maple [A]  time = 0.004, size = 43, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{2}{x}^{2}-4\,xab-8\,{b}^{2} \right ) }{3\,{a}^{3}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a*x+b)*(a^2*x^2-4*a*b*x-8*b^2)/a^3/x^(3/2)/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 0.963199, size = 74, normalized size = 1. \begin{align*} \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 6 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{3}} - \frac{2 \, b^{2}}{\sqrt{a + \frac{b}{x}} a^{3} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((a + b/x)^(3/2)*x^(3/2) - 6*sqrt(a + b/x)*b*sqrt(x))/a^3 - 2*b^2/(sqrt(a + b/x)*a^3*sqrt(x))

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Fricas [A]  time = 1.48386, size = 101, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} - 4 \, a b x - 8 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{4} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a^2*x^2 - 4*a*b*x - 8*b^2)*sqrt(x)*sqrt((a*x + b)/x)/(a^4*x + a^3*b)

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Sympy [B]  time = 4.41296, size = 206, normalized size = 2.78 \begin{align*} \frac{2 a^{3} b^{\frac{9}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{6 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{24 a b^{\frac{13}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{16 b^{\frac{15}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*b**(9/2)*x**3*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 6*a**2*b**(11/2)*x**2*
sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 24*a*b**(13/2)*x*sqrt(a*x/b + 1)/(3*a**5*b*
*4*x**2 + 6*a**4*b**5*x + 3*a**3*b**6) - 16*b**(15/2)*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a*
*3*b**6)

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Giac [A]  time = 1.16496, size = 59, normalized size = 0.8 \begin{align*} \frac{16 \, b^{\frac{3}{2}}}{3 \, a^{3}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 6 \, \sqrt{a x + b} b - \frac{3 \, b^{2}}{\sqrt{a x + b}}\right )}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/3*b^(3/2)/a^3 + 2/3*((a*x + b)^(3/2) - 6*sqrt(a*x + b)*b - 3*b^2/sqrt(a*x + b))/a^3