3.1731 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^6} \, dx\)
Optimal. Leaf size=99 \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 a^4 \sqrt{a+\frac{b}{x}}}{b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]
[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 = 0.0385148, 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, Rules used =
{266, 43} \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 a^4 \sqrt{a+\frac{b}{x}}}{b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]
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)
Rule 266
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]]
Rule 43
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])
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} x^6} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{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] time = 0.0330358, size = 62, normalized size = 0.63 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (48 a^2 b^2 x^2-64 a^3 b x^3+128 a^4 x^4-40 a b^3 x+35 b^4\right )}{315 b^5 x^4} \]
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)
________________________________________________________________________________________
Maple [A] time = 0.005, size = 66, normalized size = 0.7 \begin{align*} -{\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}}}}}} \end{align*}
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)
________________________________________________________________________________________
Maxima [A] time = 1.0023, size = 109, normalized size = 1.1 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
Fricas [A] time = 1.49072, size = 140, normalized size = 1.41 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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] time = 4.72554, size = 4901, normalized size = 49.51 \begin{align*} \text{result too large to display} \end{align*}
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) + 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))
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Giac [A] time = 1.15501, size = 161, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (315 \, a^{4} \sqrt{\frac{a x + b}{x}} - \frac{420 \,{\left (a x + b\right )} a^{3} \sqrt{\frac{a x + b}{x}}}{x} + \frac{378 \,{\left (a x + b\right )}^{2} a^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{180 \,{\left (a x + b\right )}^{3} a \sqrt{\frac{a x + b}{x}}}{x^{3}} + \frac{35 \,{\left (a x + b\right )}^{4} \sqrt{\frac{a x + b}{x}}}{x^{4}}\right )}}{315 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^6/(a+b/x)^(1/2),x, algorithm="giac")
[Out]
-2/315*(315*a^4*sqrt((a*x + b)/x) - 420*(a*x + b)*a^3*sqrt((a*x + b)/x)/x + 378*(a*x + b)^2*a^2*sqrt((a*x + b)
/x)/x^2 - 180*(a*x + b)^3*a*sqrt((a*x + b)/x)/x^3 + 35*(a*x + b)^4*sqrt((a*x + b)/x)/x^4)/b^5