\(\int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx\) [1731]

Optimal result

Integrand size = 15, antiderivative size = 99 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 a^4 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5} \]

[Out]

8/3*a^3*(a+b/x)^(3/2)/b^5-12/5*a^2*(a+b/x)^(5/2)/b^5+8/7*a*(a+b/x)^(7/2)/b^5-2/9*(a+b/x)^(9/2)/b^5-2*a^4*(a+b/
x)^(1/2)/b^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 a^4 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5} \]

[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)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a^4}{b^4 \sqrt {a+b x}}-\frac {4 a^3 \sqrt {a+b x}}{b^4}+\frac {6 a^2 (a+b x)^{3/2}}{b^4}-\frac {4 a (a+b x)^{5/2}}{b^4}+\frac {(a+b x)^{7/2}}{b^4}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\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 {8 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (35 b^4-40 a b^3 x+48 a^2 b^2 x^2-64 a^3 b x^3+128 a^4 x^4\right )}{315 b^5 x^4} \]

[In]

Integrate[1/(Sqrt[a + b/x]*x^6),x]

[Out]

(-2*Sqrt[(b + a*x)/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)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66

method	result	size
trager	\(-\frac {2 \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{315 x^{4} b^{5}}\)	\(65\)
gosper	\(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right )}{315 x^{5} b^{5} \sqrt {\frac {a x +b}{x}}}\)	\(66\)
risch	\(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right )}{315 x^{5} b^{5} \sqrt {\frac {a x +b}{x}}}\)	\(66\)
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, \left (630 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} x^{6}+630 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} x^{6}+315 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}-315 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}-1260 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{4}+748 a^{\frac {7}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,x^{3}-492 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} x^{2}+300 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x -140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}\right )}{630 x^{5} \sqrt {x \left (a x +b \right )}\, b^{6} \sqrt {a}}\)	\(241\)

[In]

int(1/x^6/(a+b/x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315/x^4*(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)/b^5*(-(-a*x-b)/x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \, {\left (128 \, a^{4} x^{4} - 64 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{315 \, b^{5} x^{4}} \]

[In]

integrate(1/x^6/(a+b/x)^(1/2),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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4901 vs. \(2 (85) = 170\).

Time = 2.48 (sec) , antiderivative size = 4901, normalized size of antiderivative = 49.51 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\text {Too large to display} \]

[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) + 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)) - 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) + 315*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**3
5*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*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(2
9/2) + 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)) - 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**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 46252*a**(27
/2)*b**(59/2)*x**9*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)) - 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) + 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)) - 91
60*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) + 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)) - 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**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)) - 2446*a**(13/2)*b**(73/2)*x**2*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3
150*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) + 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**19*b**24*x**(29/2)/(315*a**(29/2)*b**29*x**(29/2) + 3
150*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) + 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)) + 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**(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**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) + 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**(2
1/2)*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) + 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**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) + 37
800*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**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{5}} - \frac {12 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3}}{3 \, b^{5}} - \frac {2 \, \sqrt {a + \frac {b}{x}} a^{4}}{b^{5}} \]

[In]

integrate(1/x^6/(a+b/x)^(1/2),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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\frac {2 \, {\left (1008 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} + 1680 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b + 1080 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} + 35 \, b^{4}\right )}}{315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/x^6/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

2/315*(1008*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2 + 1680*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b + 1080*(s
qrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2 + 315*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3 + 35*b^4)/((sqrt(a)*x
- sqrt(a*x^2 + b*x))^9*sgn(x))

Mupad [B] (verification not implemented)

Time = 5.70 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\frac {16\,a\,\sqrt {a+\frac {b}{x}}}{63\,b^2\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x}}}{9\,b\,x^4}-\frac {256\,a^4\,\sqrt {a+\frac {b}{x}}}{315\,b^5}-\frac {32\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b^3\,x^2}+\frac {128\,a^3\,\sqrt {a+\frac {b}{x}}}{315\,b^4\,x} \]

[In]

int(1/(x^6*(a + b/x)^(1/2)),x)

[Out]

(16*a*(a + b/x)^(1/2))/(63*b^2*x^3) - (2*(a + b/x)^(1/2))/(9*b*x^4) - (256*a^4*(a + b/x)^(1/2))/(315*b^5) - (3
2*a^2*(a + b/x)^(1/2))/(105*b^3*x^2) + (128*a^3*(a + b/x)^(1/2))/(315*b^4*x)