3.1697 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^5} \, dx\)
Optimal. Leaf size=80 \[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4}-\frac{6 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4} \]
[Out]
(2*a^3*(a + b/x)^(3/2))/(3*b^4) - (6*a^2*(a + b/x)^(5/2))/(5*b^4) + (6*a*(a + b/
x)^(7/2))/(7*b^4) - (2*(a + b/x)^(9/2))/(9*b^4)
_______________________________________________________________________________________
Rubi [A] time = 0.0967987, antiderivative size = 80, 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
\[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4}-\frac{6 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x]/x^5,x]
[Out]
(2*a^3*(a + b/x)^(3/2))/(3*b^4) - (6*a^2*(a + b/x)^(5/2))/(5*b^4) + (6*a*(a + b/
x)^(7/2))/(7*b^4) - (2*(a + b/x)^(9/2))/(9*b^4)
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Rubi in Sympy [A] time = 13.1763, size = 68, normalized size = 0.85 \[ \frac{2 a^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{4}} - \frac{6 a^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{6 a \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(1/2)/x**5,x)
[Out]
2*a**3*(a + b/x)**(3/2)/(3*b**4) - 6*a**2*(a + b/x)**(5/2)/(5*b**4) + 6*a*(a + b
/x)**(7/2)/(7*b**4) - 2*(a + b/x)**(9/2)/(9*b**4)
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Mathematica [A] time = 0.03159, size = 62, normalized size = 0.78 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (16 a^4 x^4-8 a^3 b x^3+6 a^2 b^2 x^2-5 a b^3 x-35 b^4\right )}{315 b^4 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x]/x^5,x]
[Out]
(2*Sqrt[a + b/x]*(-35*b^4 - 5*a*b^3*x + 6*a^2*b^2*x^2 - 8*a^3*b*x^3 + 16*a^4*x^4
))/(315*b^4*x^4)
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Maple [A] time = 0.01, size = 55, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}-24\,{a}^{2}b{x}^{2}+30\,a{b}^{2}x-35\,{b}^{3} \right ) }{315\,{x}^{4}{b}^{4}}\sqrt{{\frac{ax+b}{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(1/2)/x^5,x)
[Out]
2/315*(a*x+b)*(16*a^3*x^3-24*a^2*b*x^2+30*a*b^2*x-35*b^3)*((a*x+b)/x)^(1/2)/x^4/
b^4
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Maxima [A] time = 1.43562, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{4}} + \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{4}} - \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^5,x, algorithm="maxima")
[Out]
-2/9*(a + b/x)^(9/2)/b^4 + 6/7*(a + b/x)^(7/2)*a/b^4 - 6/5*(a + b/x)^(5/2)*a^2/b
^4 + 2/3*(a + b/x)^(3/2)*a^3/b^4
_______________________________________________________________________________________
Fricas [A] time = 0.224562, size = 81, normalized size = 1.01 \[ \frac{2 \,{\left (16 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 5 \, a b^{3} x - 35 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^5,x, algorithm="fricas")
[Out]
2/315*(16*a^4*x^4 - 8*a^3*b*x^3 + 6*a^2*b^2*x^2 - 5*a*b^3*x - 35*b^4)*sqrt((a*x
+ b)/x)/(b^4*x^4)
_______________________________________________________________________________________
Sympy [A] time = 9.61742, size = 2297, normalized size = 28.71 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(1/2)/x**5,x)
[Out]
32*a**(29/2)*b**(23/2)*x**10*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 18
90*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b
**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2)
+ 315*a**(9/2)*b**21*x**(9/2)) + 176*a**(27/2)*b**(25/2)*x**9*sqrt(a*x/b + 1)/(
315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*
b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2
) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 396*a**(25/2
)*b**(27/2)*x**8*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)
*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/
2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9
/2)*b**21*x**(9/2)) + 462*a**(23/2)*b**(29/2)*x**7*sqrt(a*x/b + 1)/(315*a**(21/2
)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17
/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**
(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 210*a**(21/2)*b**(31/2)*
x**6*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(1
9/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a*
*(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x*
*(9/2)) - 378*a**(19/2)*b**(33/2)*x**5*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(
21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a
**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20
*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 1134*a**(17/2)*b**(35/2)*x**4*sqrt(a
*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725
*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**
19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 1
494*a**(15/2)*b**(37/2)*x**3*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 18
90*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b
**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2)
+ 315*a**(9/2)*b**21*x**(9/2)) - 1098*a**(13/2)*b**(39/2)*x**2*sqrt(a*x/b + 1)/
(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)
*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/
2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 430*a**(11/
2)*b**(41/2)*x*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b
**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2)
+ 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2
)*b**21*x**(9/2)) - 70*a**(9/2)*b**(43/2)*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x
**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 630
0*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b*
*20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 32*a**15*b**11*x**(21/2)/(315*a**
(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x
**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 189
0*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 192*a**14*b**12*x**
(19/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a*
*(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*
x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 480*
a**13*b**13*x**(17/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(
19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a
**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x
**(9/2)) - 640*a**12*b**14*x**(15/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(1
9/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**
(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a
**(9/2)*b**21*x**(9/2)) - 480*a**11*b**15*x**(13/2)/(315*a**(21/2)*b**15*x**(21/
2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(
15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x*
*(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 192*a**10*b**16*x**(11/2)/(315*a**(21/2
)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17
/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**
(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 32*a**9*b**17*x**(9/2)/(
315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*
b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2
) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.263627, size = 239, normalized size = 2.99 \[ \frac{2 \,{\left (630 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}}{\rm sign}\left (x\right ) + 1764 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b{\rm sign}\left (x\right ) + 1995 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{2}{\rm sign}\left (x\right ) + 1125 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{3}{\rm sign}\left (x\right ) + 315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{4}{\rm sign}\left (x\right ) + 35 \, b^{5}{\rm sign}\left (x\right )\right )}}{315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^5,x, algorithm="giac")
[Out]
2/315*(630*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*sign(x) + 1764*(sqrt(a)*x -
sqrt(a*x^2 + b*x))^4*a^2*b*sign(x) + 1995*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(
3/2)*b^2*sign(x) + 1125*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^3*sign(x) + 315*(s
qrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^4*sign(x) + 35*b^5*sign(x))/(sqrt(a)*x -
sqrt(a*x^2 + b*x))^9