3.1721 \(\int \frac{(a+\frac{b}{x})^{5/2}}{x^6} \, dx\)
Optimal. Leaf size=101 \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{2 a^4 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{15/2}}{15 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5} \]
[Out]
(-2*a^4*(a + b/x)^(7/2))/(7*b^5) + (8*a^3*(a + b/x)^(9/2))/(9*b^5) - (12*a^2*(a + b/x)^(11/2))/(11*b^5) + (8*a
*(a + b/x)^(13/2))/(13*b^5) - (2*(a + b/x)^(15/2))/(15*b^5)
________________________________________________________________________________________
Rubi [A] time = 0.0395268, 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} \[ -\frac{12 a^2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{2 a^4 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{15/2}}{15 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b/x)^(5/2)/x^6,x]
[Out]
(-2*a^4*(a + b/x)^(7/2))/(7*b^5) + (8*a^3*(a + b/x)^(9/2))/(9*b^5) - (12*a^2*(a + b/x)^(11/2))/(11*b^5) + (8*a
*(a + b/x)^(13/2))/(13*b^5) - (2*(a + b/x)^(15/2))/(15*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{\left (a+\frac{b}{x}\right )^{5/2}}{x^6} \, dx &=-\operatorname{Subst}\left (\int x^4 (a+b x)^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^4 (a+b x)^{5/2}}{b^4}-\frac{4 a^3 (a+b x)^{7/2}}{b^4}+\frac{6 a^2 (a+b x)^{9/2}}{b^4}-\frac{4 a (a+b x)^{11/2}}{b^4}+\frac{(a+b x)^{13/2}}{b^4}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^4 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{15/2}}{15 b^5}\\ \end{align*}
Mathematica [A] time = 0.0320573, size = 69, normalized size = 0.68 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (1008 a^2 b^2 x^2-448 a^3 b x^3+128 a^4 x^4-1848 a b^3 x+3003 b^4\right )}{45045 b^5 x^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b/x)^(5/2)/x^6,x]
[Out]
(-2*Sqrt[a + b/x]*(b + a*x)^3*(3003*b^4 - 1848*a*b^3*x + 1008*a^2*b^2*x^2 - 448*a^3*b*x^3 + 128*a^4*x^4))/(450
45*b^5*x^7)
________________________________________________________________________________________
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}-448\,{a}^{3}{x}^{3}b+1008\,{a}^{2}{x}^{2}{b}^{2}-1848\,ax{b}^{3}+3003\,{b}^{4} \right ) }{45045\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b/x)^(5/2)/x^6,x)
[Out]
-2/45045*(a*x+b)*(128*a^4*x^4-448*a^3*b*x^3+1008*a^2*b^2*x^2-1848*a*b^3*x+3003*b^4)*((a*x+b)/x)^(5/2)/x^5/b^5
________________________________________________________________________________________
Maxima [A] time = 1.19532, size = 109, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{15}{2}}}{15 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}} a}{13 \, b^{5}} - \frac{12 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} a^{2}}{11 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a^{3}}{9 \, b^{5}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{4}}{7 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(5/2)/x^6,x, algorithm="maxima")
[Out]
-2/15*(a + b/x)^(15/2)/b^5 + 8/13*(a + b/x)^(13/2)*a/b^5 - 12/11*(a + b/x)^(11/2)*a^2/b^5 + 8/9*(a + b/x)^(9/2
)*a^3/b^5 - 2/7*(a + b/x)^(7/2)*a^4/b^5
________________________________________________________________________________________
Fricas [A] time = 1.67725, size = 220, normalized size = 2.18 \begin{align*} -\frac{2 \,{\left (128 \, a^{7} x^{7} - 64 \, a^{6} b x^{6} + 48 \, a^{5} b^{2} x^{5} - 40 \, a^{4} b^{3} x^{4} + 35 \, a^{3} b^{4} x^{3} + 4473 \, a^{2} b^{5} x^{2} + 7161 \, a b^{6} x + 3003 \, b^{7}\right )} \sqrt{\frac{a x + b}{x}}}{45045 \, b^{5} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(5/2)/x^6,x, algorithm="fricas")
[Out]
-2/45045*(128*a^7*x^7 - 64*a^6*b*x^6 + 48*a^5*b^2*x^5 - 40*a^4*b^3*x^4 + 35*a^3*b^4*x^3 + 4473*a^2*b^5*x^2 + 7
161*a*b^6*x + 3003*b^7)*sqrt((a*x + b)/x)/(b^5*x^7)
________________________________________________________________________________________
Sympy [B] time = 6.53156, size = 5482, normalized size = 54.28 \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)**(5/2)/x**6,x)
[Out]
-256*a**(49/2)*b**(49/2)*x**17*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(3
3/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/
2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2
) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) -
2432*a**(47/2)*b**(51/2)*x**16*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(3
3/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/
2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2
) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) -
10336*a**(45/2)*b**(53/2)*x**15*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(
33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27
/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/
2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) -
25840*a**(43/2)*b**(55/2)*x**14*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**
(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(2
7/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21
/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2))
- 41990*a**(41/2)*b**(57/2)*x**13*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x*
*(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(
27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(2
1/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2))
- 55198*a**(39/2)*b**(59/2)*x**12*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x
**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**
(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(
21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)
) - 138996*a**(37/2)*b**(61/2)*x**11*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30
*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x
**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x*
*(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/
2)) - 571428*a**(35/2)*b**(63/2)*x**10*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**
30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33
*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*
x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(1
5/2)) - 1788930*a**(33/2)*b**(65/2)*x**9*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b
**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**
33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**3
6*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**
(15/2)) - 3876730*a**(31/2)*b**(67/2)*x**8*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)
*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b
**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b*
*36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x
**(15/2)) - 5991128*a**(29/2)*b**(69/2)*x**7*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/
2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)
*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*
b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39
*x**(15/2)) - 6754696*a**(27/2)*b**(71/2)*x**6*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(3
3/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/
2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2
)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**
39*x**(15/2)) - 5597098*a**(25/2)*b**(73/2)*x**5*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**
(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(2
7/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21
/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b
**39*x**(15/2)) - 3383090*a**(23/2)*b**(75/2)*x**4*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a
**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**
(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(
21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)
*b**39*x**(15/2)) - 1454740*a**(21/2)*b**(77/2)*x**3*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450
*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a
**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a*
*(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/
2)*b**39*x**(15/2)) - 422436*a**(19/2)*b**(79/2)*x**2*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 45045
0*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*
a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a
**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15
/2)*b**39*x**(15/2)) - 74382*a**(17/2)*b**(81/2)*x*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a
**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**
(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(
21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)
*b**39*x**(15/2)) - 6006*a**(15/2)*b**(83/2)*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/
2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)
*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*
b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39
*x**(15/2)) + 256*a**25*b**24*x**(35/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) +
2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11
351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 202
7025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 2560*a*
*24*b**25*x**(33/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b*
*31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**
34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37
*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 11520*a**23*b**26*x**(31/2)
/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 540
5400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459
450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450
*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 30720*a**22*b**27*x**(29/2)/(45045*a**(35/2)*b
**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**3
2*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35
*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x*
*(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 53760*a**21*b**28*x**(27/2)/(45045*a**(35/2)*b**29*x**(35/2) + 45
0450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 94594
50*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 540540
0*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**
(15/2)*b**39*x**(15/2)) + 64512*a**20*b**29*x**(25/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**3
0*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*
x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x
**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15
/2)) + 53760*a**19*b**30*x**(23/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 20270
25*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 1135134
0*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*
a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 30720*a**18*
b**31*x**(21/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*
x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x
**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**
(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 11520*a**17*b**32*x**(19/2)/(45
045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400
*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*
a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**
(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 2560*a**16*b**33*x**(17/2)/(45045*a**(35/2)*b**29*
x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**
(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(
23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/
2) + 45045*a**(15/2)*b**39*x**(15/2)) + 256*a**15*b**34*x**(15/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a*
*(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(
27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(2
1/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*
b**39*x**(15/2))
________________________________________________________________________________________
Giac [B] time = 1.2324, size = 448, normalized size = 4.44 \begin{align*} \frac{2 \,{\left (144144 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{10} a^{5} \mathrm{sgn}\left (x\right ) + 960960 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9} a^{\frac{9}{2}} b \mathrm{sgn}\left (x\right ) + 2934360 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4} b^{2} \mathrm{sgn}\left (x\right ) + 5360355 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 6451445 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{4} \mathrm{sgn}\left (x\right ) + 5324319 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{5} \mathrm{sgn}\left (x\right ) + 3042585 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{6} \mathrm{sgn}\left (x\right ) + 1186185 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{7} \mathrm{sgn}\left (x\right ) + 301455 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{8} \mathrm{sgn}\left (x\right ) + 45045 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{9} \mathrm{sgn}\left (x\right ) + 3003 \, b^{10} \mathrm{sgn}\left (x\right )\right )}}{45045 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x)^(5/2)/x^6,x, algorithm="giac")
[Out]
2/45045*(144144*(sqrt(a)*x - sqrt(a*x^2 + b*x))^10*a^5*sgn(x) + 960960*(sqrt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/
2)*b*sgn(x) + 2934360*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*b^2*sgn(x) + 5360355*(sqrt(a)*x - sqrt(a*x^2 + b*x
))^7*a^(7/2)*b^3*sgn(x) + 6451445*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^4*sgn(x) + 5324319*(sqrt(a)*x - sqrt
(a*x^2 + b*x))^5*a^(5/2)*b^5*sgn(x) + 3042585*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^6*sgn(x) + 1186185*(sqrt
(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^7*sgn(x) + 301455*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^8*sgn(x) + 4504
5*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^9*sgn(x) + 3003*b^10*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^15