∖int∘ ___ a+ b x ________

3.1698 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^6} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 a^4 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5} \]

[Out]

(-2*a^4*(a + b/x)^(3/2))/(3*b^5) + (8*a^3*(a + b/x)^(5/2))/(5*b^5) - (12*a^2*(a

+ b/x)^(7/2))/(7*b^5) + (8*a*(a + b/x)^(9/2))/(9*b^5) - (2*(a + b/x)^(11/2))/(11

*b^5)

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Rubi [A] time = 0.116262, antiderivative size = 101, 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 \[ -\frac{2 a^4 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5} \]

Antiderivative was successfully veriﬁed.

[In] Int[Sqrt[a + b/x]/x^6,x]

[Out]

(-2*a^4*(a + b/x)^(3/2))/(3*b^5) + (8*a^3*(a + b/x)^(5/2))/(5*b^5) - (12*a^2*(a

+ b/x)^(7/2))/(7*b^5) + (8*a*(a + b/x)^(9/2))/(9*b^5) - (2*(a + b/x)^(11/2))/(11

*b^5)

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Rubi in Sympy [A] time = 16.0787, size = 87, normalized size = 0.86 \[ - \frac{2 a^{4} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{5}} + \frac{8 a^{3} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{5}} - \frac{12 a^{2} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{5}} + \frac{8 a \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{5}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((a+b/x)**(1/2)/x**6,x)

[Out]

-2*a**4*(a + b/x)**(3/2)/(3*b**5) + 8*a**3*(a + b/x)**(5/2)/(5*b**5) - 12*a**2*(

a + b/x)**(7/2)/(7*b**5) + 8*a*(a + b/x)**(9/2)/(9*b**5) - 2*(a + b/x)**(11/2)/(

11*b**5)

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Mathematica [A] time = 0.0348282, size = 73, normalized size = 0.72 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (128 a^5 x^5-64 a^4 b x^4+48 a^3 b^2 x^3-40 a^2 b^3 x^2+35 a b^4 x+315 b^5\right )}{3465 b^5 x^5} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[Sqrt[a + b/x]/x^6,x]

[Out]

(-2*Sqrt[a + b/x]*(315*b^5 + 35*a*b^4*x - 40*a^2*b^3*x^2 + 48*a^3*b^2*x^3 - 64*a

^4*b*x^4 + 128*a^5*x^5))/(3465*b^5*x^5)

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Maple [A] time = 0.009, size = 66, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}-192\,{a}^{3}{x}^{3}b+240\,{a}^{2}{x}^{2}{b}^{2}-280\,ax{b}^{3}+315\,{b}^{4} \right ) }{3465\,{x}^{5}{b}^{5}}\sqrt{{\frac{ax+b}{x}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x)^(1/2)/x^6,x)

[Out]

-2/3465*(a*x+b)*(128*a^4*x^4-192*a^3*b*x^3+240*a^2*b^2*x^2-280*a*b^3*x+315*b^4)*

((a*x+b)/x)^(1/2)/x^5/b^5

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Maxima [A] time = 1.43809, size = 109, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}}}{11 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a}{9 \, b^{5}} - \frac{12 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{3}}{5 \, b^{5}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{4}}{3 \, b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(a + b/x)/x^6,x, algorithm="maxima")

[Out]

-2/11*(a + b/x)^(11/2)/b^5 + 8/9*(a + b/x)^(9/2)*a/b^5 - 12/7*(a + b/x)^(7/2)*a^

2/b^5 + 8/5*(a + b/x)^(5/2)*a^3/b^5 - 2/3*(a + b/x)^(3/2)*a^4/b^5

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Fricas [A] time = 0.222849, size = 96, normalized size = 0.95 \[ -\frac{2 \,{\left (128 \, a^{5} x^{5} - 64 \, a^{4} b x^{4} + 48 \, a^{3} b^{2} x^{3} - 40 \, a^{2} b^{3} x^{2} + 35 \, a b^{4} x + 315 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{3465 \, b^{5} x^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(a + b/x)/x^6,x, algorithm="fricas")

[Out]

-2/3465*(128*a^5*x^5 - 64*a^4*b*x^4 + 48*a^3*b^2*x^3 - 40*a^2*b^3*x^2 + 35*a*b^4

*x + 315*b^5)*sqrt((a*x + b)/x)/(b^5*x^5)

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Sympy [A] time = 16.7957, size = 5095, normalized size = 50.45 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a+b/x)**(1/2)/x**6,x)

[Out]

-256*a**(41/2)*b**(49/2)*x**15*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) +

34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**

(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**

34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/

2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a

**(11/2)*b**39*x**(11/2)) - 2432*a**(39/2)*b**(51/2)*x**14*sqrt(a*x/b + 1)/(3465

*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*

b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(

23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 41

5800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13

/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 10336*a**(37/2)*b**(53/2

)*x**13*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*

x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2)

+ 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a

**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b

**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2

)) - 25840*a**(35/2)*b**(55/2)*x**12*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(3

1/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 4158

00*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/

2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x

**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) +

3465*a**(11/2)*b**39*x**(11/2)) - 41990*a**(33/2)*b**(57/2)*x**11*sqrt(a*x/b + 1

)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**

(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**

33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/

2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650

*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 46882*a**(31/2)*b

**(59/2)*x**10*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)

*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**

(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 7

27650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(

15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x

**(11/2)) - 41514*a**(29/2)*b**(61/2)*x**9*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29

*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2)

+ 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a

**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b

**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13

/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 47982*a**(27/2)*b**(63/2)*x**8*sqrt(a*x/

b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 15592

5*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2

)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x*

*(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) +

34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 86460*a**(25

/2)*b**(65/2)*x**7*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(2

9/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32

*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2)

+ 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*

a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**

39*x**(11/2)) - 141460*a**(23/2)*b**(67/2)*x**6*sqrt(a*x/b + 1)/(3465*a**(31/2)*

b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(2

7/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873

180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17

/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x

**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 167156*a**(21/2)*b**(69/2)*x**5*sqr

t(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) +

155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a*

*(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b*

*35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15

/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 137932

*a**(19/2)*b**(71/2)*x**4*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 3465

0*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2

)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x*

*(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) +

155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11

/2)*b**39*x**(11/2)) - 78046*a**(17/2)*b**(73/2)*x**3*sqrt(a*x/b + 1)/(3465*a**(

31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31

*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2)

+ 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*

a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b

**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 28970*a**(15/2)*b**(75/2)*x**

2*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29

/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 7276

50*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/

2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x

**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 6

370*a**(13/2)*b**(77/2)*x*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 3465

0*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2

)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x*

*(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) +

155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11

/2)*b**39*x**(11/2)) - 630*a**(11/2)*b**(79/2)*sqrt(a*x/b + 1)/(3465*a**(31/2)*b

**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27

/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 8731

80*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/

2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x*

*(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 256*a**21*b**24*x**(31/2)/(3465*a**(

31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31

*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2)

+ 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*

a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b

**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 2560*a**20*b**25*x**(29/2)/(3

465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/

2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x

**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) +

415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**

(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 11520*a**19*b**26*x**

(27/2)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 15592

5*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2

)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x*

*(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) +

34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 30720*a**18*

b**27*x**(25/2)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2

) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650

*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)

*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**

(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 537

60*a**17*b**28*x**(23/2)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30

*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2)

+ 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*

a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*

b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/

2)) + 64512*a**16*b**29*x**(21/2)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29

/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*

x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2)

+ 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 155925*a

**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**3

9*x**(11/2)) + 53760*a**15*b**30*x**(19/2)/(3465*a**(31/2)*b**29*x**(31/2) + 346

50*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/

2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x

**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) +

155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a**(1

1/2)*b**39*x**(11/2)) + 30720*a**14*b**31*x**(17/2)/(3465*a**(31/2)*b**29*x**(31

/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 41580

0*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2

)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x*

*(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3

465*a**(11/2)*b**39*x**(11/2)) + 11520*a**13*b**32*x**(15/2)/(3465*a**(31/2)*b**

29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2

) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180

*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)

*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(

13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 2560*a**12*b**33*x**(13/2)/(3465*a**(3

1/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*

x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2)

+ 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a

**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b*

*38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) + 256*a**11*b**34*x**(11/2)/(346

5*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)

*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**

(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 4

15800*a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(1

3/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.263888, size = 281, normalized size = 2.78 \[ \frac{2 \,{\left (11088 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3}{\rm sign}\left (x\right ) + 36960 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b{\rm sign}\left (x\right ) + 51480 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2}{\rm sign}\left (x\right ) + 38115 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) + 15785 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4}{\rm sign}\left (x\right ) + 3465 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5}{\rm sign}\left (x\right ) + 315 \, b^{6}{\rm sign}\left (x\right )\right )}}{3465 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(a + b/x)/x^6,x, algorithm="giac")

[Out]

2/3465*(11088*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sign(x) + 36960*(sqrt(a)*x -

sqrt(a*x^2 + b*x))^5*a^(5/2)*b*sign(x) + 51480*(sqrt(a)*x - sqrt(a*x^2 + b*x))^

4*a^2*b^2*sign(x) + 38115*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^3*sign(x)

+ 15785*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sign(x) + 3465*(sqrt(a)*x - sqrt

(a*x^2 + b*x))*sqrt(a)*b^5*sign(x) + 315*b^6*sign(x))/(sqrt(a)*x - sqrt(a*x^2 +

b*x))^11

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