3.1700 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^6} \, dx\)
Optimal. Leaf size=101 \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 a^4 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.0425936, 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, Rules used =
{266, 43} \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 a^4 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5} \]
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)
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{\sqrt{a+\frac{b}{x}}}{x^6} \, dx &=-\operatorname{Subst}\left (\int x^4 \sqrt{a+b x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^4 \sqrt{a+b x}}{b^4}-\frac{4 a^3 (a+b x)^{3/2}}{b^4}+\frac{6 a^2 (a+b x)^{5/2}}{b^4}-\frac{4 a (a+b x)^{7/2}}{b^4}+\frac{(a+b x)^{9/2}}{b^4}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\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{8 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}\\ \end{align*}
Mathematica [A] time = 0.0304776, size = 67, normalized size = 0.66 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b) \left (240 a^2 b^2 x^2-192 a^3 b x^3+128 a^4 x^4-280 a b^3 x+315 b^4\right )}{3465 b^5 x^5} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b/x]/x^6,x]
[Out]
(-2*Sqrt[a + b/x]*(b + a*x)*(315*b^4 - 280*a*b^3*x + 240*a^2*b^2*x^2 - 192*a^3*b*x^3 + 128*a^4*x^4))/(3465*b^5
*x^5)
________________________________________________________________________________________
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}-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}}}} \end{align*}
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
________________________________________________________________________________________
Maxima [A] time = 0.9759, size = 109, normalized size = 1.08 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(1/2)/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
________________________________________________________________________________________
Fricas [A] time = 1.79957, size = 166, normalized size = 1.64 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(1/2)/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)
________________________________________________________________________________________
Sympy [B] time = 6.08754, size = 5095, normalized size = 50.45 \begin{align*} \text{result too large to display} \end{align*}
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) + 15592
5*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**(2
1/2)*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)) - 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**(1
5/2) + 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*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) + 3465
0*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) + 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) + 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) + 727650*a**(19/2)*b**3
5*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**(2
5/2) + 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) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 72765
0*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**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(2
3/2) + 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)) - 16715
6*a**(21/2)*b**(69/2)*x**5*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)) - 137932*a**(19/2)
*b**(71/2)*x**4*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)) - 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) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 7276
50*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)) - 6370*a**(13/2)*b**(77/2)*x*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)) - 630*a**(11/2)*b**(79/2)*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) + 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) + 4158
00*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)) + 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) + 15
5925*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**2
0*b**25*x**(29/2)/(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)) + 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) + 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)) + 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**3
6*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)) + 53760*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**(2
1/2)*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) + 415
800*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)) + 53760*a**15*b**30*x**(19/2)/(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) + 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) + 4158
00*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) + 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
**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**3
1*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)/(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**3
6*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))
________________________________________________________________________________________
Giac [B] time = 1.17444, size = 281, normalized size = 2.78 \begin{align*} \frac{2 \,{\left (11088 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} \mathrm{sgn}\left (x\right ) + 36960 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b \mathrm{sgn}\left (x\right ) + 51480 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm{sgn}\left (x\right ) + 38115 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 15785 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4} \mathrm{sgn}\left (x\right ) + 3465 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5} \mathrm{sgn}\left (x\right ) + 315 \, b^{6} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(1/2)/x^6,x, algorithm="giac")
[Out]
2/3465*(11088*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sgn(x) + 36960*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b
*sgn(x) + 51480*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^2*sgn(x) + 38115*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(
3/2)*b^3*sgn(x) + 15785*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sgn(x) + 3465*(sqrt(a)*x - sqrt(a*x^2 + b*x))*
sqrt(a)*b^5*sgn(x) + 315*b^6*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^11