3.1784 \(\int \frac{x^{5/2}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 b^4}{35 a^5 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}+\frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(-256*b^4)/(35*a^5*Sqrt[a + b/x]*Sqrt[x]) - (128*b^3*Sqrt[x])/(35*a^4*Sqrt[a + b
/x]) + (32*b^2*x^(3/2))/(35*a^3*Sqrt[a + b/x]) - (16*b*x^(5/2))/(35*a^2*Sqrt[a +
b/x]) + (2*x^(7/2))/(7*a*Sqrt[a + b/x])

_______________________________________________________________________________________

Rubi [A]  time = 0.15138, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{256 b^4}{35 a^5 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}+\frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b/x)^(3/2),x]

[Out]

(-256*b^4)/(35*a^5*Sqrt[a + b/x]*Sqrt[x]) - (128*b^3*Sqrt[x])/(35*a^4*Sqrt[a + b
/x]) + (32*b^2*x^(3/2))/(35*a^3*Sqrt[a + b/x]) - (16*b*x^(5/2))/(35*a^2*Sqrt[a +
b/x]) + (2*x^(7/2))/(7*a*Sqrt[a + b/x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4756, size = 110, normalized size = 0.87 \[ \frac{2 x^{\frac{7}{2}}}{7 a \sqrt{a + \frac{b}{x}}} - \frac{16 b x^{\frac{5}{2}}}{35 a^{2} \sqrt{a + \frac{b}{x}}} + \frac{32 b^{2} x^{\frac{3}{2}}}{35 a^{3} \sqrt{a + \frac{b}{x}}} - \frac{128 b^{3} \sqrt{x}}{35 a^{4} \sqrt{a + \frac{b}{x}}} - \frac{256 b^{4}}{35 a^{5} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(a+b/x)**(3/2),x)

[Out]

2*x**(7/2)/(7*a*sqrt(a + b/x)) - 16*b*x**(5/2)/(35*a**2*sqrt(a + b/x)) + 32*b**2
*x**(3/2)/(35*a**3*sqrt(a + b/x)) - 128*b**3*sqrt(x)/(35*a**4*sqrt(a + b/x)) - 2
56*b**4/(35*a**5*sqrt(x)*sqrt(a + b/x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0622255, size = 71, normalized size = 0.56 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (5 a^4 x^4-8 a^3 b x^3+16 a^2 b^2 x^2-64 a b^3 x-128 b^4\right )}{35 a^5 (a x+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(a + b/x)^(3/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-128*b^4 - 64*a*b^3*x + 16*a^2*b^2*x^2 - 8*a^3*b*x^3 +
5*a^4*x^4))/(35*a^5*(b + a*x))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 66, normalized size = 0.5 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 5\,{x}^{4}{a}^{4}-8\,b{x}^{3}{a}^{3}+16\,{b}^{2}{x}^{2}{a}^{2}-64\,{b}^{3}xa-128\,{b}^{4} \right ) }{35\,{a}^{5}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(a+b/x)^(3/2),x)

[Out]

2/35*(a*x+b)*(5*a^4*x^4-8*a^3*b*x^3+16*a^2*b^2*x^2-64*a*b^3*x-128*b^4)/a^5/x^(3/
2)/((a*x+b)/x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.44638, size = 122, normalized size = 0.97 \[ -\frac{2 \, b^{4}}{\sqrt{a + \frac{b}{x}} a^{5} \sqrt{x}} + \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 28 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 70 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}} - 140 \, \sqrt{a + \frac{b}{x}} b^{3} \sqrt{x}\right )}}{35 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

-2*b^4/(sqrt(a + b/x)*a^5*sqrt(x)) + 2/35*(5*(a + b/x)^(7/2)*x^(7/2) - 28*(a + b
/x)^(5/2)*b*x^(5/2) + 70*(a + b/x)^(3/2)*b^2*x^(3/2) - 140*sqrt(a + b/x)*b^3*sqr
t(x))/a^5

_______________________________________________________________________________________

Fricas [A]  time = 0.234759, size = 81, normalized size = 0.64 \[ \frac{2 \,{\left (5 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x - 128 \, b^{4}\right )}}{35 \, a^{5} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(a + b/x)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*a^4*x^4 - 8*a^3*b*x^3 + 16*a^2*b^2*x^2 - 64*a*b^3*x - 128*b^4)/(a^5*sqrt
(x)*sqrt((a*x + b)/x))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(a+b/x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231839, size = 95, normalized size = 0.75 \[ \frac{256 \, b^{\frac{7}{2}}}{35 \, a^{5}} + \frac{2 \,{\left (5 \,{\left (a x + b\right )}^{\frac{7}{2}} - 28 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 140 \, \sqrt{a x + b} b^{3} - \frac{35 \, b^{4}}{\sqrt{a x + b}}\right )}}{35 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(a + b/x)^(3/2),x, algorithm="giac")

[Out]

256/35*b^(7/2)/a^5 + 2/35*(5*(a*x + b)^(7/2) - 28*(a*x + b)^(5/2)*b + 70*(a*x +
b)^(3/2)*b^2 - 140*sqrt(a*x + b)*b^3 - 35*b^4/sqrt(a*x + b))/a^5