3.1746 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^4} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0782579, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.91689, size = 46, normalized size = 0.84 \[ \frac{2 a^{2}}{3 b^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{4 a}{b^{3} \sqrt{a + \frac{b}{x}}} - \frac{2 \sqrt{a + \frac{b}{x}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.040834, size = 44, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+12 a b x+3 b^2\right )}{3 b^3 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+12\,abx+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43613, size = 63, normalized size = 1.15 \[ -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b^{3}} - \frac{4 \, a}{\sqrt{a + \frac{b}{x}} b^{3}} + \frac{2 \, a^{2}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^4),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.238412, size = 65, normalized size = 1.18 \[ -\frac{2 \,{\left (8 \, a^{2} x^{2} + 12 \, a b x + 3 \, b^{2}\right )}}{3 \,{\left (a b^{3} x^{2} + b^{4} x\right )} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 14.0104, size = 136, normalized size = 2.47 \[ \begin{cases} - \frac{16 a^{2} x^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{24 a b x}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{6 b^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{5}{2}} x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2)/x**4,x)

[Out]

Piecewise((-16*a**2*x**2/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x))
- 24*a*b*x/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)) - 6*b**2/(3*a*
b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)), Ne(b, 0)), (-1/(3*a**(5/2)*x*
*3), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270123, size = 78, normalized size = 1.42 \[ \frac{2}{3} \, b{\left (\frac{{\left (a^{2} - \frac{6 \,{\left (a x + b\right )} a}{x}\right )} x}{{\left (a x + b\right )} b^{4} \sqrt{\frac{a x + b}{x}}} - \frac{3 \, \sqrt{\frac{a x + b}{x}}}{b^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^4),x, algorithm="giac")

[Out]

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