3.1796 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} \sqrt{x}} \, dx\)

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 = 0.0821112, 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 \[ \frac{16 b^2}{3 a^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{8 b}{a^2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 \sqrt{x}}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

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))

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Rubi in Sympy [A]  time = 6.84131, size = 60, normalized size = 0.86 \[ \frac{2 \sqrt{x}}{a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{8 b}{a^{2} \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{16 b^{2}}{3 a^{3} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x)/(a*(a + b/x)**(3/2)) + 8*b/(a**2*sqrt(x)*(a + b/x)**(3/2)) + 16*b**2/(
3*a**3*x**(3/2)*(a + b/x)**(3/2))

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Mathematica [A]  time = 0.0517752, 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 = 0.009, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{a}^{2}{x}^{2}+12\,abx+8\,{b}^{2} \right ) }{3\,{a}^{3}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(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 = 1.43223, size = 70, normalized size = 1. \[ \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a^{3}} + \frac{2 \,{\left (6 \,{\left (a + \frac{b}{x}\right )} b x - b^{2}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*sqrt(x)),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 = 0.231567, size = 65, normalized size = 0.93 \[ \frac{2 \,{\left (3 \, a^{2} x^{2} + 12 \, a b x + 8 \, b^{2}\right )}}{3 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*sqrt(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 171.696, size = 151, normalized size = 2.16 \[ \frac{6 a^{2} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} + \frac{24 a b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} + \frac{16 b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \]

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/XCAS [A]  time = 0.233449, size = 62, normalized size = 0.89 \[ \frac{2 \,{\left (3 \, \sqrt{a x + b} + \frac{6 \,{\left (a x + b\right )} b - b^{2}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, a^{3}} - \frac{16 \, \sqrt{b}}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*sqrt(x)),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