3.1740 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^6} \, dx\)
Optimal. Leaf size=95 \[ \frac{2 a^4}{b^5 \sqrt{a+\frac{b}{x}}}+\frac{8 a^3 \sqrt{a+\frac{b}{x}}}{b^5}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]
[Out]
(2*a^4)/(b^5*Sqrt[a + b/x]) + (8*a^3*Sqrt[a + b/x])/b^5 - (4*a^2*(a + b/x)^(3/2))/b^5 + (8*a*(a + b/x)^(5/2))/
(5*b^5) - (2*(a + b/x)^(7/2))/(7*b^5)
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Rubi [A] time = 0.039751, antiderivative size = 95, 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} \[ \frac{2 a^4}{b^5 \sqrt{a+\frac{b}{x}}}+\frac{8 a^3 \sqrt{a+\frac{b}{x}}}{b^5}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b/x)^(3/2)*x^6),x]
[Out]
(2*a^4)/(b^5*Sqrt[a + b/x]) + (8*a^3*Sqrt[a + b/x])/b^5 - (4*a^2*(a + b/x)^(3/2))/b^5 + (8*a*(a + b/x)^(5/2))/
(5*b^5) - (2*(a + b/x)^(7/2))/(7*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}{\left (a+\frac{b}{x}\right )^{3/2} x^6} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{3/2}}-\frac{4 a^3}{b^4 \sqrt{a+b x}}+\frac{6 a^2 \sqrt{a+b x}}{b^4}-\frac{4 a (a+b x)^{3/2}}{b^4}+\frac{(a+b x)^{5/2}}{b^4}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^4}{b^5 \sqrt{a+\frac{b}{x}}}+\frac{8 a^3 \sqrt{a+\frac{b}{x}}}{b^5}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}\\ \end{align*}
Mathematica [A] time = 0.0335074, size = 62, normalized size = 0.65 \[ \frac{2 \left (-16 a^2 b^2 x^2+64 a^3 b x^3+128 a^4 x^4+8 a b^3 x-5 b^4\right )}{35 b^5 x^4 \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b/x)^(3/2)*x^6),x]
[Out]
(2*(-5*b^4 + 8*a*b^3*x - 16*a^2*b^2*x^2 + 64*a^3*b*x^3 + 128*a^4*x^4))/(35*b^5*Sqrt[a + b/x]*x^4)
________________________________________________________________________________________
Maple [A] time = 0.006, 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-16\,{a}^{2}{x}^{2}{b}^{2}+8\,ax{b}^{3}-5\,{b}^{4} \right ) }{35\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x)^(3/2)/x^6,x)
[Out]
2/35*(a*x+b)*(128*a^4*x^4+64*a^3*b*x^3-16*a^2*b^2*x^2+8*a*b^3*x-5*b^4)/x^5/b^5/((a*x+b)/x)^(3/2)
________________________________________________________________________________________
Maxima [A] time = 1.00182, size = 109, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{5 \, b^{5}} - \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2}}{b^{5}} + \frac{8 \, \sqrt{a + \frac{b}{x}} a^{3}}{b^{5}} + \frac{2 \, a^{4}}{\sqrt{a + \frac{b}{x}} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="maxima")
[Out]
-2/7*(a + b/x)^(7/2)/b^5 + 8/5*(a + b/x)^(5/2)*a/b^5 - 4*(a + b/x)^(3/2)*a^2/b^5 + 8*sqrt(a + b/x)*a^3/b^5 + 2
*a^4/(sqrt(a + b/x)*b^5)
________________________________________________________________________________________
Fricas [A] time = 1.59202, size = 151, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (128 \, a^{4} x^{4} + 64 \, a^{3} b x^{3} - 16 \, a^{2} b^{2} x^{2} + 8 \, a b^{3} x - 5 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{35 \,{\left (a b^{5} x^{4} + b^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="fricas")
[Out]
2/35*(128*a^4*x^4 + 64*a^3*b*x^3 - 16*a^2*b^2*x^2 + 8*a*b^3*x - 5*b^4)*sqrt((a*x + b)/x)/(a*b^5*x^4 + b^6*x^3)
________________________________________________________________________________________
Sympy [B] time = 6.92584, size = 4707, normalized size = 49.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)**(3/2)/x**6,x)
[Out]
256*a**(33/2)*b**(49/2)*x**13*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) +
1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17
/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x
**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 2432*a**(31/2)*b**(51/2)*x**12*sqrt(a*x
/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*
a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b
**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2)
+ 35*a**(7/2)*b**39*x**(7/2)) + 10336*a**(29/2)*b**(53/2)*x**11*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2)
+ 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(
19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36
*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 2584
0*a**(27/2)*b**(55/2)*x**10*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 15
75*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2
)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**
(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 41990*a**(25/2)*b**(57/2)*x**9*sqrt(a*x/b
+ 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a*
*(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**
35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) +
35*a**(7/2)*b**39*x**(7/2)) + 46182*a**(23/2)*b**(59/2)*x**8*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) +
350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/
2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x*
*(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 34584*a
**(21/2)*b**(61/2)*x**7*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a
**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b*
*34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/
2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 17112*a**(19/2)*b**(63/2)*x**6*sqrt(a*x/b + 1
)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21
/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x
**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*
a**(7/2)*b**39*x**(7/2)) + 4980*a**(17/2)*b**(65/2)*x**5*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a
**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b*
*33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/
2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 340*a**(15/2
)*b**(67/2)*x**4*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2
)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**
(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 35
0*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 424*a**(13/2)*b**(69/2)*x**3*sqrt(a*x/b + 1)/(35*a**
(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32
*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2)
+ 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*
b**39*x**(7/2)) - 248*a**(11/2)*b**(71/2)*x**2*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b
**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19
/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*
a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 74*a**(9/2)*b**(73/2)*
x*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/
2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a
**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**3
8*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 10*a**(7/2)*b**(75/2)*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2)
+ 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(
19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36
*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 256*
a**17*b**24*x**(27/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**
(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 73
50*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*
b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 2560*a**16*b**25*x**(25/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*
a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b
**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13
/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 11520*a**15
*b**26*x**(23/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2
) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a*
*(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38
*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 30720*a**14*b**27*x**(21/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(
25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33
*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2)
+ 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 53760*a**13*b**
28*x**(19/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) +
4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15
/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**
(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 64512*a**12*b**29*x**(17/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2
)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**
(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 15
75*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 53760*a**11*b**30*x
**(15/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200
*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*
b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2
) + 35*a**(7/2)*b**39*x**(7/2)) - 30720*a**10*b**31*x**(13/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b*
*30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/
2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a
**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 11520*a**9*b**32*x**(11
/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(
21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35
*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 3
5*a**(7/2)*b**39*x**(7/2)) - 2560*a**8*b**33*x**(9/2)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(
25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 882
0*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)
*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) - 256*a**7*b**34*x**(7/2)/(35*a**
(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32
*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2)
+ 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*
b**39*x**(7/2))
________________________________________________________________________________________
Giac [A] time = 1.25753, size = 171, normalized size = 1.8 \begin{align*} \frac{2}{35} \, b{\left (\frac{35 \, a^{4}}{b^{6} \sqrt{\frac{a x + b}{x}}} + \frac{140 \, a^{3} b^{36} \sqrt{\frac{a x + b}{x}} - \frac{70 \,{\left (a x + b\right )} a^{2} b^{36} \sqrt{\frac{a x + b}{x}}}{x} + \frac{28 \,{\left (a x + b\right )}^{2} a b^{36} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{5 \,{\left (a x + b\right )}^{3} b^{36} \sqrt{\frac{a x + b}{x}}}{x^{3}}}{b^{42}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="giac")
[Out]
2/35*b*(35*a^4/(b^6*sqrt((a*x + b)/x)) + (140*a^3*b^36*sqrt((a*x + b)/x) - 70*(a*x + b)*a^2*b^36*sqrt((a*x + b
)/x)/x + 28*(a*x + b)^2*a*b^36*sqrt((a*x + b)/x)/x^2 - 5*(a*x + b)^3*b^36*sqrt((a*x + b)/x)/x^3)/b^42)