3.1696 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3} \]

[Out]

(-2*a^2*(a + b/x)^(3/2))/(3*b^3) + (4*a*(a + b/x)^(5/2))/(5*b^3) - (2*(a + b/x)^
(7/2))/(7*b^3)

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Rubi [A]  time = 0.0779898, antiderivative size = 59, 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 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(3/2))/(3*b^3) + (4*a*(a + b/x)^(5/2))/(5*b^3) - (2*(a + b/x)^
(7/2))/(7*b^3)

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Rubi in Sympy [A]  time = 9.79602, size = 49, normalized size = 0.83 \[ - \frac{2 a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{4 a \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(a + b/x)**(3/2)/(3*b**3) + 4*a*(a + b/x)**(5/2)/(5*b**3) - 2*(a + b/x)*
*(7/2)/(7*b**3)

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Mathematica [A]  time = 0.0246777, size = 51, normalized size = 0.86 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (8 a^3 x^3-4 a^2 b x^2+3 a b^2 x+15 b^3\right )}{105 b^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/x^4,x]

[Out]

(-2*Sqrt[a + b/x]*(15*b^3 + 3*a*b^2*x - 4*a^2*b*x^2 + 8*a^3*x^3))/(105*b^3*x^3)

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Maple [A]  time = 0.007, size = 44, normalized size = 0.8 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-12\,abx+15\,{b}^{2} \right ) }{105\,{b}^{3}{x}^{3}}\sqrt{{\frac{ax+b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^4,x)

[Out]

-2/105*(a*x+b)*(8*a^2*x^2-12*a*b*x+15*b^2)*((a*x+b)/x)^(1/2)/b^3/x^3

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Maxima [A]  time = 1.44091, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{5 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^4,x, algorithm="maxima")

[Out]

-2/7*(a + b/x)^(7/2)/b^3 + 4/5*(a + b/x)^(5/2)*a/b^3 - 2/3*(a + b/x)^(3/2)*a^2/b
^3

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Fricas [A]  time = 0.225879, size = 66, normalized size = 1.12 \[ -\frac{2 \,{\left (8 \, a^{3} x^{3} - 4 \, a^{2} b x^{2} + 3 \, a b^{2} x + 15 \, b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{105 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^4,x, algorithm="fricas")

[Out]

-2/105*(8*a^3*x^3 - 4*a^2*b*x^2 + 3*a*b^2*x + 15*b^3)*sqrt((a*x + b)/x)/(b^3*x^3
)

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Sympy [A]  time = 6.48161, size = 899, normalized size = 15.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(19/2)*b**(9/2)*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*
a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7
/2)) - 40*a**(17/2)*b**(11/2)*x**5*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2)
+ 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**1
0*x**(7/2)) - 30*a**(15/2)*b**(13/2)*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x*
*(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/
2)*b**10*x**(7/2)) - 40*a**(13/2)*b**(15/2)*x**3*sqrt(a*x/b + 1)/(105*a**(13/2)*
b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105
*a**(7/2)*b**10*x**(7/2)) - 100*a**(11/2)*b**(17/2)*x**2*sqrt(a*x/b + 1)/(105*a*
*(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/
2) + 105*a**(7/2)*b**10*x**(7/2)) - 96*a**(9/2)*b**(19/2)*x*sqrt(a*x/b + 1)/(105
*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**
(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 30*a**(7/2)*b**(21/2)*sqrt(a*x/b + 1)/(10
5*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x*
*(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16*a**10*b**4*x**(13/2)/(105*a**(13/2)*b
**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*
a**(7/2)*b**10*x**(7/2)) + 48*a**9*b**5*x**(11/2)/(105*a**(13/2)*b**7*x**(13/2)
+ 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10
*x**(7/2)) + 48*a**8*b**6*x**(9/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)
*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16
*a**7*b**7*x**(7/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2)
+ 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2))

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GIAC/XCAS [A]  time = 0.256476, size = 197, normalized size = 3.34 \[ \frac{2 \,{\left (140 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2}{\rm sign}\left (x\right ) + 315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b{\rm sign}\left (x\right ) + 273 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{2}{\rm sign}\left (x\right ) + 105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{3}{\rm sign}\left (x\right ) + 15 \, b^{4}{\rm sign}\left (x\right )\right )}}{105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^4,x, algorithm="giac")

[Out]

2/105*(140*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*sign(x) + 315*(sqrt(a)*x - sqrt
(a*x^2 + b*x))^3*a^(3/2)*b*sign(x) + 273*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2
*sign(x) + 105*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*sign(x) + 15*b^4*sign
(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^7