3.1797 \(\int \frac{\sqrt{x}}{(a+\frac{b}{x})^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{32 b^3}{3 a^4 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b^2}{a^3 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-32*b^3)/(3*a^4*(a + b/x)^(3/2)*x^(3/2)) - (16*b^2)/(a^3*(a + b/x)^(3/2)*Sqrt[x]) - (4*b*Sqrt[x])/(a^2*(a + b
/x)^(3/2)) + (2*x^(3/2))/(3*a*(a + b/x)^(3/2))

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Rubi [A]  time = 0.0349012, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ -\frac{32 b^3}{3 a^4 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b^2}{a^3 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b/x)^(5/2),x]

[Out]

(-32*b^3)/(3*a^4*(a + b/x)^(3/2)*x^(3/2)) - (16*b^2)/(a^3*(a + b/x)^(3/2)*Sqrt[x]) - (4*b*Sqrt[x])/(a^2*(a + b
/x)^(3/2)) + (2*x^(3/2))/(3*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{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{(2 b) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} \sqrt{x}} \, dx}{a}\\ &=-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 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^{3/2}} \, dx}{a^2}\\ &=-\frac{16 b^2}{a^3 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}}+\frac{\left (16 b^3\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{5/2}} \, dx}{a^3}\\ &=-\frac{32 b^3}{3 a^4 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}-\frac{16 b^2}{a^3 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0177735, size = 59, normalized size = 0.61 \[ \frac{2 \left (-6 a^2 b x^2+a^3 x^3-24 a b^2 x-16 b^3\right )}{3 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b/x)^(5/2),x]

[Out]

(2*(-16*b^3 - 24*a*b^2*x - 6*a^2*b*x^2 + a^3*x^3))/(3*a^4*Sqrt[a + b/x]*Sqrt[x]*(b + a*x))

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Maple [A]  time = 0.006, size = 54, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{3}{x}^{3}-6\,{a}^{2}b{x}^{2}-24\,xa{b}^{2}-16\,{b}^{3} \right ) }{3\,{a}^{4}}{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(x^(1/2)/(a+b/x)^(5/2),x)

[Out]

2/3*(a*x+b)*(a^3*x^3-6*a^2*b*x^2-24*a*b^2*x-16*b^3)/a^4/x^(5/2)/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 0.957994, size = 96, normalized size = 1. \begin{align*} \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 9 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{4}} - \frac{2 \,{\left (9 \,{\left (a + \frac{b}{x}\right )} b^{2} x - b^{3}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{4} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((a + b/x)^(3/2)*x^(3/2) - 9*sqrt(a + b/x)*b*sqrt(x))/a^4 - 2/3*(9*(a + b/x)*b^2*x - b^3)/((a + b/x)^(3/2)
*a^4*x^(3/2))

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Fricas [A]  time = 1.45544, size = 147, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} - 6 \, a^{2} b x^{2} - 24 \, a b^{2} x - 16 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a^3*x^3 - 6*a^2*b*x^2 - 24*a*b^2*x - 16*b^3)*sqrt(x)*sqrt((a*x + b)/x)/(a^6*x^2 + 2*a^5*b*x + a^4*b^2)

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Sympy [B]  time = 15.0703, size = 320, normalized size = 3.33 \begin{align*} \frac{2 a^{4} b^{\frac{19}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{10 a^{3} b^{\frac{21}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{60 a^{2} b^{\frac{23}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{80 a b^{\frac{25}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{32 b^{\frac{27}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(a+b/x)**(5/2),x)

[Out]

2*a**4*b**(19/2)*x**4*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12) -
10*a**3*b**(21/2)*x**3*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12)
- 60*a**2*b**(23/2)*x**2*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**1
2) - 80*a*b**(25/2)*x*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12) -
32*b**(27/2)*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12)

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Giac [A]  time = 1.25759, size = 77, normalized size = 0.8 \begin{align*} \frac{32 \, b^{\frac{3}{2}}}{3 \, a^{4}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{a x + b} b - \frac{9 \,{\left (a x + b\right )} b^{2} - b^{3}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/3*b^(3/2)/a^4 + 2/3*((a*x + b)^(3/2) - 9*sqrt(a*x + b)*b - (9*(a*x + b)*b^2 - b^3)/(a*x + b)^(3/2))/a^4