3.1798 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} \sqrt{x}} \, dx\)

Optimal. Leaf size=70 \[ \frac{16 b^2}{3 a^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{8 b}{a^2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0223814, antiderivative size = 70, 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} \[ \frac{16 b^2}{3 a^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{8 b}{a^2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*Sqrt[x]),x]

[Out]

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

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 )^{5/2} \sqrt{x}} \, dx &=\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{(4 b) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{3/2}} \, dx}{a}\\ &=\frac{8 b}{a^2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}+\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{\left (8 b^2\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{5/2}} \, dx}{a^2}\\ &=\frac{16 b^2}{3 a^3 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}+\frac{8 b}{a^2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}+\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0284906, size = 49, normalized size = 0.7 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+12 a b x+8 b^2\right )}{3 a^3 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*Sqrt[x]),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(8*b^2 + 12*a*b*x + 3*a^2*x^2))/(3*a^3*(b + a*x)^2)

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Maple [A]  time = 0.005, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{a}^{2}{x}^{2}+12\,xab+8\,{b}^{2} \right ) }{3\,{a}^{3}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^(1/2),x)

[Out]

2/3*(a*x+b)*(3*a^2*x^2+12*a*b*x+8*b^2)/a^3/x^(5/2)/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 0.958935, size = 70, normalized size = 1. \begin{align*} \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a^{3}} + \frac{2 \,{\left (6 \,{\left (a + \frac{b}{x}\right )} b x - b^{2}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49743, size = 127, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (3 \, a^{2} x^{2} + 12 \, a b x + 8 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(1/2),x, algorithm="fricas")

[Out]

2/3*(3*a^2*x^2 + 12*a*b*x + 8*b^2)*sqrt(x)*sqrt((a*x + b)/x)/(a^5*x^2 + 2*a^4*b*x + a^3*b^2)

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Sympy [B]  time = 25.0123, size = 151, normalized size = 2.16 \begin{align*} \frac{6 a^{2} b^{\frac{9}{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{11}{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{13}{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(1/(a+b/x)**(5/2)/x**(1/2),x)

[Out]

6*a**2*b**(9/2)*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**(11/2)*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**(13/2)*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 = 1.94862, size = 62, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{a x + b} + \frac{6 \,{\left (a x + b\right )} b - b^{2}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, a^{3}} - \frac{16 \, \sqrt{b}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(1/2),x, algorithm="giac")

[Out]

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