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3.1729 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^6} \, dx\)

Optimal. Leaf size=99 \[ -\frac{2 a^4 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]

[Out]

(-2*a^4*Sqrt[a + b/x])/b^5 + (8*a^3*(a + b/x)^(3/2))/(3*b^5) - (12*a^2*(a + b/x)
^(5/2))/(5*b^5) + (8*a*(a + b/x)^(7/2))/(7*b^5) - (2*(a + b/x)^(9/2))/(9*b^5)

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Rubi [A]  time = 0.11411, antiderivative size = 99, 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 \[ -\frac{2 a^4 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^6),x]

[Out]

(-2*a^4*Sqrt[a + b/x])/b^5 + (8*a^3*(a + b/x)^(3/2))/(3*b^5) - (12*a^2*(a + b/x)
^(5/2))/(5*b^5) + (8*a*(a + b/x)^(7/2))/(7*b^5) - (2*(a + b/x)^(9/2))/(9*b^5)

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Rubi in Sympy [A]  time = 16.4331, size = 85, normalized size = 0.86 \[ - \frac{2 a^{4} \sqrt{a + \frac{b}{x}}}{b^{5}} + \frac{8 a^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{5}} - \frac{12 a^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{5}} + \frac{8 a \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{5}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**4*sqrt(a + b/x)/b**5 + 8*a**3*(a + b/x)**(3/2)/(3*b**5) - 12*a**2*(a + b/x
)**(5/2)/(5*b**5) + 8*a*(a + b/x)**(7/2)/(7*b**5) - 2*(a + b/x)**(9/2)/(9*b**5)

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Mathematica [A]  time = 0.0433638, size = 62, normalized size = 0.63 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (128 a^4 x^4-64 a^3 b x^3+48 a^2 b^2 x^2-40 a b^3 x+35 b^4\right )}{315 b^5 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x^6),x]

[Out]

(-2*Sqrt[a + b/x]*(35*b^4 - 40*a*b^3*x + 48*a^2*b^2*x^2 - 64*a^3*b*x^3 + 128*a^4
*x^4))/(315*b^5*x^4)

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Maple [A]  time = 0.008, size = 66, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}-64\,{a}^{3}{x}^{3}b+48\,{a}^{2}{x}^{2}{b}^{2}-40\,ax{b}^{3}+35\,{b}^{4} \right ) }{315\,{x}^{5}{b}^{5}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(a+b/x)^(1/2),x)

[Out]

-2/315*(a*x+b)*(128*a^4*x^4-64*a^3*b*x^3+48*a^2*b^2*x^2-40*a*b^3*x+35*b^4)/x^5/b
^5/((a*x+b)/x)^(1/2)

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Maxima [A]  time = 1.44645, size = 109, normalized size = 1.1 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{5}} - \frac{12 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{5}} - \frac{2 \, \sqrt{a + \frac{b}{x}} a^{4}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^6),x, algorithm="maxima")

[Out]

-2/9*(a + b/x)^(9/2)/b^5 + 8/7*(a + b/x)^(7/2)*a/b^5 - 12/5*(a + b/x)^(5/2)*a^2/
b^5 + 8/3*(a + b/x)^(3/2)*a^3/b^5 - 2*sqrt(a + b/x)*a^4/b^5

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Fricas [A]  time = 0.221999, size = 81, normalized size = 0.82 \[ -\frac{2 \,{\left (128 \, a^{4} x^{4} - 64 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^6),x, algorithm="fricas")

[Out]

-2/315*(128*a^4*x^4 - 64*a^3*b*x^3 + 48*a^2*b^2*x^2 - 40*a*b^3*x + 35*b^4)*sqrt(
(a*x + b)/x)/(b^5*x^4)

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Sympy [A]  time = 17.8442, size = 4901, normalized size = 49.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256*a**(37/2)*b**(49/2)*x**14*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) +
3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/
2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**
(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 141
75*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**
39*x**(9/2)) - 2432*a**(35/2)*b**(51/2)*x**13*sqrt(a*x/b + 1)/(315*a**(29/2)*b**
29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2)
+ 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(
19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*
x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 3
15*a**(9/2)*b**39*x**(9/2)) - 10336*a**(33/2)*b**(53/2)*x**12*sqrt(a*x/b + 1)/(3
15*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*
b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21
/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*
a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**
38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 25840*a**(31/2)*b**(55/2)*x**11*sq
rt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) +
14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21
/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x*
*(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 31
50*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 41990*a**(29/2)*b*
*(57/2)*x**10*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b*
*30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2
) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a*
*(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**3
7*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 46
252*a**(27/2)*b**(59/2)*x**9*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 31
50*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)
*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(1
9/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175
*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39
*x**(9/2)) - 35214*a**(25/2)*b**(61/2)*x**8*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29
*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) +
37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19
/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x*
*(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315
*a**(9/2)*b**39*x**(9/2)) - 19632*a**(23/2)*b**(63/2)*x**7*sqrt(a*x/b + 1)/(315*
a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**
31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2)
 + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**
(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*
x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 10860*a**(21/2)*b**(65/2)*x**6*sqrt(a
*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 1417
5*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*
b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17
/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a
**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 9160*a**(19/2)*b**(67/
2)*x**5*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x*
*(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66
150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2
)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(
13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 8396*a**
(17/2)*b**(69/2)*x**4*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(
27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*
x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) +
66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13
/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/
2)) - 5632*a**(15/2)*b**(71/2)*x**3*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/
2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a*
*(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**3
4*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2)
+ 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2
)*b**39*x**(9/2)) - 2446*a**(13/2)*b**(73/2)*x**2*sqrt(a*x/b + 1)/(315*a**(29/2)
*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25
/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*
a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b*
*36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2)
 + 315*a**(9/2)*b**39*x**(9/2)) - 620*a**(11/2)*b**(75/2)*x*sqrt(a*x/b + 1)/(315
*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b*
*31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2
) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a*
*(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38
*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 70*a**(9/2)*b**(77/2)*sqrt(a*x/b + 1
)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25
/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x*
*(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37
800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)
*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) + 256*a**19*b**24*x**(29/2)/(315
*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b*
*31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2
) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a*
*(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38
*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) + 2560*a**18*b**25*x**(27/2)/(315*a**(
29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x
**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 7
9380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/
2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(
11/2) + 315*a**(9/2)*b**39*x**(9/2)) + 11520*a**17*b**26*x**(25/2)/(315*a**(29/2
)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(2
5/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380
*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b
**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2
) + 315*a**(9/2)*b**39*x**(9/2)) + 30720*a**16*b**27*x**(23/2)/(315*a**(29/2)*b*
*29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2)
 + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**
(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36
*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) +
315*a**(9/2)*b**39*x**(9/2)) + 53760*a**15*b**28*x**(21/2)/(315*a**(29/2)*b**29*
x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 3
7800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/
2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**
(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*
a**(9/2)*b**39*x**(9/2)) + 64512*a**14*b**29*x**(19/2)/(315*a**(29/2)*b**29*x**(
29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800
*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b
**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/
2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(
9/2)*b**39*x**(9/2)) + 53760*a**13*b**30*x**(17/2)/(315*a**(29/2)*b**29*x**(29/2
) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**
(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34
*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) +
 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)
*b**39*x**(9/2)) + 30720*a**12*b**31*x**(15/2)/(315*a**(29/2)*b**29*x**(29/2) +
3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/
2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**
(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 141
75*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**
39*x**(9/2)) + 11520*a**11*b**32*x**(13/2)/(315*a**(29/2)*b**29*x**(29/2) + 3150
*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b
**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/
2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a
**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x
**(9/2)) + 2560*a**10*b**33*x**(11/2)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(
27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*
x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) +
66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13
/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/
2)) + 256*a**9*b**34*x**(9/2)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b*
*30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2
) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a*
*(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**3
7*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.254545, size = 181, normalized size = 1.83 \[ -\frac{2 \,{\left (315 \, a^{4} b^{48} \sqrt{\frac{a x + b}{x}} - \frac{420 \,{\left (a x + b\right )} a^{3} b^{48} \sqrt{\frac{a x + b}{x}}}{x} + \frac{378 \,{\left (a x + b\right )}^{2} a^{2} b^{48} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{180 \,{\left (a x + b\right )}^{3} a b^{48} \sqrt{\frac{a x + b}{x}}}{x^{3}} + \frac{35 \,{\left (a x + b\right )}^{4} b^{48} \sqrt{\frac{a x + b}{x}}}{x^{4}}\right )}}{315 \, b^{53}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^6),x, algorithm="giac")

[Out]

-2/315*(315*a^4*b^48*sqrt((a*x + b)/x) - 420*(a*x + b)*a^3*b^48*sqrt((a*x + b)/x
)/x + 378*(a*x + b)^2*a^2*b^48*sqrt((a*x + b)/x)/x^2 - 180*(a*x + b)^3*a*b^48*sq
rt((a*x + b)/x)/x^3 + 35*(a*x + b)^4*b^48*sqrt((a*x + b)/x)/x^4)/b^53

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