3.1747 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^5} \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 a^3}{3 b^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+\frac{b}{x}}}+\frac{6 a \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0967667, antiderivative size = 76, 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 \[ -\frac{2 a^3}{3 b^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+\frac{b}{x}}}+\frac{6 a \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0037, size = 65, normalized size = 0.86 \[ - \frac{2 a^{3}}{3 b^{4} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{6 a^{2}}{b^{4} \sqrt{a + \frac{b}{x}}} + \frac{6 a \sqrt{a + \frac{b}{x}}}{b^{4}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0474628, size = 58, normalized size = 0.76 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (16 a^3 x^3+24 a^2 b x^2+6 a b^2 x-b^3\right )}{3 b^4 x (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 55, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}+24\,{a}^{2}b{x}^{2}+6\,a{b}^{2}x-{b}^{3} \right ) }{3\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.43812, size = 86, normalized size = 1.13 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{4}} + \frac{6 \, \sqrt{a + \frac{b}{x}} a}{b^{4}} + \frac{6 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{4}} - \frac{2 \, a^{3}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234988, size = 82, normalized size = 1.08 \[ \frac{2 \,{\left (16 \, a^{3} x^{3} + 24 \, a^{2} b x^{2} + 6 \, a b^{2} x - b^{3}\right )}}{3 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.3151, size = 187, normalized size = 2.46 \[ \begin{cases} \frac{32 a^{3} x^{3}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} + \frac{48 a^{2} b x^{2}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} + \frac{12 a b^{2} x}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} - \frac{2 b^{3}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{5}{2}} x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((32*a**3*x**3/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)
) + 48*a**2*b*x**2/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)) + 1
2*a*b**2*x/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)) - 2*b**3/(3
*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)), Ne(b, 0)), (-1/(4*a**(5
/2)*x**4), True))

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GIAC/XCAS [A]  time = 0.261653, size = 123, normalized size = 1.62 \[ -\frac{2}{3} \, b{\left (\frac{{\left (a^{3} - \frac{9 \,{\left (a x + b\right )} a^{2}}{x}\right )} x}{{\left (a x + b\right )} b^{5} \sqrt{\frac{a x + b}{x}}} - \frac{9 \, a b^{10} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{10} \sqrt{\frac{a x + b}{x}}}{x}}{b^{15}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*b*((a^3 - 9*(a*x + b)*a^2/x)*x/((a*x + b)*b^5*sqrt((a*x + b)/x)) - (9*a*b^1
0*sqrt((a*x + b)/x) - (a*x + b)*b^10*sqrt((a*x + b)/x)/x)/b^15)