3.1795 \(\int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx\)

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 = 0.114544, 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 \[ -\frac{32 b^3}{3 a^4 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b^2}{a^3 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b \sqrt{x}}{a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{3/2}}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

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))

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Rubi in Sympy [A]  time = 9.84762, size = 83, normalized size = 0.86 \[ \frac{2 x^{\frac{3}{2}}}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{4 b \sqrt{x}}{a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{16 b^{2}}{a^{3} \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{32 b^{3}}{3 a^{4} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(a+b/x)**(5/2),x)

[Out]

2*x**(3/2)/(3*a*(a + b/x)**(3/2)) - 4*b*sqrt(x)/(a**2*(a + b/x)**(3/2)) - 16*b**
2/(a**3*sqrt(x)*(a + b/x)**(3/2)) - 32*b**3/(3*a**4*x**(3/2)*(a + b/x)**(3/2))

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Mathematica [A]  time = 0.057692, size = 59, normalized size = 0.61 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (a^3 x^3-6 a^2 b x^2-24 a b^2 x-16 b^3\right )}{3 a^4 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 54, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{3}{x}^{3}-6\,{a}^{2}b{x}^{2}-24\,a{b}^{2}x-16\,{b}^{3} \right ) }{3\,{a}^{4}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

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 = 1.43585, size = 96, normalized size = 1. \[ \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 9 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{4}} - \frac{2 \,{\left (9 \,{\left (a + \frac{b}{x}\right )} b^{2} x - b^{3}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{4} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(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 = 0.23434, size = 78, normalized size = 0.81 \[ \frac{2 \,{\left (a^{3} x^{3} - 6 \, a^{2} b x^{2} - 24 \, a b^{2} x - 16 \, b^{3}\right )}}{3 \,{\left (a^{5} x + a^{4} b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(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)/((a^5*x + a^4*b)*sqrt(x)*sqrt(
(a*x + b)/x))

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Sympy [A]  time = 90.0592, size = 320, normalized size = 3.33 \[ \frac{2 a^{4} b^{\frac{19}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{10 a^{3} b^{\frac{21}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{60 a^{2} b^{\frac{23}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{80 a b^{\frac{25}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac{32 b^{\frac{27}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} \]

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**12) - 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/XCAS [A]  time = 0.234062, size = 77, normalized size = 0.8 \[ \frac{32 \, b^{\frac{3}{2}}}{3 \, a^{4}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{a x + b} b - \frac{9 \,{\left (a x + b\right )} b^{2} - b^{3}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(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