3.810 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^9} \, dx\)

Optimal. Leaf size=126 \[ \frac{c^3 (b c-a d)}{d^5 \sqrt{c+\frac{d}{x^2}}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5} \]

[Out]

(c^3*(b*c - a*d))/(d^5*Sqrt[c + d/x^2]) + (c^2*(4*b*c - 3*a*d)*Sqrt[c + d/x^2])/
d^5 - (c*(2*b*c - a*d)*(c + d/x^2)^(3/2))/d^5 + ((4*b*c - a*d)*(c + d/x^2)^(5/2)
)/(5*d^5) - (b*(c + d/x^2)^(7/2))/(7*d^5)

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Rubi [A]  time = 0.290172, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c^3 (b c-a d)}{d^5 \sqrt{c+\frac{d}{x^2}}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]

[Out]

(c^3*(b*c - a*d))/(d^5*Sqrt[c + d/x^2]) + (c^2*(4*b*c - 3*a*d)*Sqrt[c + d/x^2])/
d^5 - (c*(2*b*c - a*d)*(c + d/x^2)^(3/2))/d^5 + ((4*b*c - a*d)*(c + d/x^2)^(5/2)
)/(5*d^5) - (b*(c + d/x^2)^(7/2))/(7*d^5)

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Rubi in Sympy [A]  time = 28.5105, size = 114, normalized size = 0.9 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7 d^{5}} - \frac{c^{3} \left (a d - b c\right )}{d^{5} \sqrt{c + \frac{d}{x^{2}}}} - \frac{c^{2} \sqrt{c + \frac{d}{x^{2}}} \left (3 a d - 4 b c\right )}{d^{5}} + \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 2 b c\right )}{d^{5}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (a d - 4 b c\right )}{5 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**9,x)

[Out]

-b*(c + d/x**2)**(7/2)/(7*d**5) - c**3*(a*d - b*c)/(d**5*sqrt(c + d/x**2)) - c**
2*sqrt(c + d/x**2)*(3*a*d - 4*b*c)/d**5 + c*(c + d/x**2)**(3/2)*(a*d - 2*b*c)/d*
*5 - (c + d/x**2)**(5/2)*(a*d - 4*b*c)/(5*d**5)

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Mathematica [A]  time = 0.120138, size = 104, normalized size = 0.83 \[ \frac{b \left (128 c^4 x^8+64 c^3 d x^6-16 c^2 d^2 x^4+8 c d^3 x^2-5 d^4\right )-7 a d x^2 \left (16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right )}{35 d^5 x^8 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]

[Out]

(-7*a*d*x^2*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6) + b*(-5*d^4 + 8*c*d^3
*x^2 - 16*c^2*d^2*x^4 + 64*c^3*d*x^6 + 128*c^4*x^8))/(35*d^5*Sqrt[c + d/x^2]*x^8
)

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Maple [A]  time = 0.011, size = 118, normalized size = 0.9 \[ -{\frac{ \left ( 112\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}+56\,a{c}^{2}{d}^{2}{x}^{6}-64\,b{c}^{3}d{x}^{6}-14\,ac{d}^{3}{x}^{4}+16\,b{c}^{2}{d}^{2}{x}^{4}+7\,a{d}^{4}{x}^{2}-8\,bc{d}^{3}{x}^{2}+5\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{35\,{d}^{5}{x}^{10}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x)

[Out]

-1/35*(112*a*c^3*d*x^8-128*b*c^4*x^8+56*a*c^2*d^2*x^6-64*b*c^3*d*x^6-14*a*c*d^3*
x^4+16*b*c^2*d^2*x^4+7*a*d^4*x^2-8*b*c*d^3*x^2+5*b*d^4)*(c*x^2+d)/((c*x^2+d)/x^2
)^(3/2)/d^5/x^10

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Maxima [A]  time = 1.3907, size = 204, normalized size = 1.62 \[ -\frac{1}{35} \, b{\left (\frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{5}} - \frac{28 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{5}} + \frac{70 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{5}} - \frac{140 \, \sqrt{c + \frac{d}{x^{2}}} c^{3}}{d^{5}} - \frac{35 \, c^{4}}{\sqrt{c + \frac{d}{x^{2}}} d^{5}}\right )} - \frac{1}{5} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{4}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{4}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{4}} + \frac{5 \, c^{3}}{\sqrt{c + \frac{d}{x^{2}}} d^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x, algorithm="maxima")

[Out]

-1/35*b*(5*(c + d/x^2)^(7/2)/d^5 - 28*(c + d/x^2)^(5/2)*c/d^5 + 70*(c + d/x^2)^(
3/2)*c^2/d^5 - 140*sqrt(c + d/x^2)*c^3/d^5 - 35*c^4/(sqrt(c + d/x^2)*d^5)) - 1/5
*a*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*sqrt(c + d/x^2)*c^2/d
^4 + 5*c^3/(sqrt(c + d/x^2)*d^4))

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Fricas [A]  time = 0.309144, size = 163, normalized size = 1.29 \[ \frac{{\left (16 \,{\left (8 \, b c^{4} - 7 \, a c^{3} d\right )} x^{8} + 8 \,{\left (8 \, b c^{3} d - 7 \, a c^{2} d^{2}\right )} x^{6} - 5 \, b d^{4} - 2 \,{\left (8 \, b c^{2} d^{2} - 7 \, a c d^{3}\right )} x^{4} +{\left (8 \, b c d^{3} - 7 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \,{\left (c d^{5} x^{8} + d^{6} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x, algorithm="fricas")

[Out]

1/35*(16*(8*b*c^4 - 7*a*c^3*d)*x^8 + 8*(8*b*c^3*d - 7*a*c^2*d^2)*x^6 - 5*b*d^4 -
2*(8*b*c^2*d^2 - 7*a*c*d^3)*x^4 + (8*b*c*d^3 - 7*a*d^4)*x^2)*sqrt((c*x^2 + d)/x
^2)/(c*d^5*x^8 + d^6*x^6)

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Sympy [A]  time = 93.8743, size = 6096, normalized size = 48.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**9,x)

[Out]

a*(-16*c**(21/2)*d**(23/2)*x**16*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 3
0*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 +
75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 88*c
**(19/2)*d**(25/2)*x**14*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15
/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(
9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 198*c**(17/2
)*d**(27/2)*x**12*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**
16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d*
*19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 231*c**(15/2)*d**(2
9/2)*x**10*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**1
5 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**
9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 145*c**(13/2)*d**(31/2)*x*
*8*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c
**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c
**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 46*c**(11/2)*d**(33/2)*x**6*sqrt(c
*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*
d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d
**20*x**7 + 5*c**(5/2)*d**21*x**5) - 8*c**(9/2)*d**(35/2)*x**4*sqrt(c*x**2/d + 1
)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13
+ 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 +
5*c**(5/2)*d**21*x**5) - 3*c**(7/2)*d**(37/2)*x**2*sqrt(c*x**2/d + 1)/(5*c**(17
/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(
11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)
*d**21*x**5) - c**(5/2)*d**(39/2)*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 +
30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11
+ 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 16*
c**11*d**11*x**17/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(1
3/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7
/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 96*c**10*d**12*x**15/(5*c**(17/2)*d**1
5*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d*
*18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x
**5) + 240*c**9*d**13*x**13/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15
+ 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9
+ 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 320*c**8*d**14*x**11/(5*c**(
17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c*
*(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/
2)*d**21*x**5) + 240*c**7*d**15*x**9/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**
16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d*
*19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 96*c**6*d**16*x**7/
(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 +
100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5
*c**(5/2)*d**21*x**5) + 16*c**5*d**17*x**5/(5*c**(17/2)*d**15*x**17 + 30*c**(15/
2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9
/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5)) + b*(128*c**(3
3/2)*d**(49/2)*x**26*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2
)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c
**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 +
4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x*
*9 + 35*c**(7/2)*d**39*x**7) + 1216*c**(31/2)*d**(51/2)*x**24*sqrt(c*x**2/d + 1)
/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x*
*23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d
**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(
11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) + 5168*c**
(29/2)*d**(53/2)*x**22*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25
/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350
*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15
+ 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*
x**9 + 35*c**(7/2)*d**39*x**7) + 12920*c**(27/2)*d**(55/2)*x**20*sqrt(c*x**2/d +
1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31
*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2
)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c
**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) + 20995
*c**(25/2)*d**(57/2)*x**18*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c*
*(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 +
7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x
**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d*
*38*x**9 + 35*c**(7/2)*d**39*x**7) + 23091*c**(23/2)*d**(59/2)*x**16*sqrt(c*x**2
/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d
**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(
17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 15
75*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) + 1
7292*c**(21/2)*d**(61/2)*x**14*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 35
0*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**2
1 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**
35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2
)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) + 8556*c**(19/2)*d**(63/2)*x**12*sqrt(c*x
**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2
)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c
**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 +
1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7)
+ 2490*c**(17/2)*d**(65/2)*x**10*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 +
350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x*
*21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d
**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9
/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) + 170*c**(15/2)*d**(67/2)*x**8*sqrt(c*x
**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2
)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c
**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 +
1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7)
- 212*c**(13/2)*d**(69/2)*x**6*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 35
0*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**2
1 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**
35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2
)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 124*c**(11/2)*d**(71/2)*x**4*sqrt(c*x**
2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*
d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**
(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1
575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) -
37*c**(9/2)*d**(73/2)*x**2*sqrt(c*x**2/d + 1)/(35*c**(27/2)*d**29*x**27 + 350*c*
*(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 +
7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x
**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d*
*38*x**9 + 35*c**(7/2)*d**39*x**7) - 5*c**(7/2)*d**(75/2)*sqrt(c*x**2/d + 1)/(35
*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23
+ 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34
*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2
)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 128*c**17*d*
*24*x**27/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)
*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c*
*(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 +
1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) -
1280*c**16*d**25*x**25/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 +
1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x
**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*
d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)
*d**39*x**7) - 5760*c**15*d**26*x**23/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*
d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**
(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4
200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9
+ 35*c**(7/2)*d**39*x**7) - 15360*c**14*d**27*x**21/(35*c**(27/2)*d**29*x**27 +
350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x
**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*
d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(
9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 26880*c**13*d**28*x**19/(35*c**(27/2
)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c*
*(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 +
7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x
**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 32256*c**12*d**29*x**
17/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*
x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)
*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c*
*(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 26880*
c**11*d**30*x**15/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c
**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 +
8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*
x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39
*x**7) - 15360*c**10*d**31*x**13/(35*c**(27/2)*d**29*x**27 + 350*c**(25/2)*d**30
*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 + 7350*c**(19/2
)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x**15 + 4200*c
**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d**38*x**9 + 35
*c**(7/2)*d**39*x**7) - 5760*c**9*d**32*x**11/(35*c**(27/2)*d**29*x**27 + 350*c*
*(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d**32*x**21 +
7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(15/2)*d**35*x
**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350*c**(9/2)*d*
*38*x**9 + 35*c**(7/2)*d**39*x**7) - 1280*c**8*d**33*x**9/(35*c**(27/2)*d**29*x*
*27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c**(21/2)*d*
*32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 + 7350*c**(1
5/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*x**11 + 350
*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7) - 128*c**7*d**34*x**7/(35*c**(27/
2)*d**29*x**27 + 350*c**(25/2)*d**30*x**25 + 1575*c**(23/2)*d**31*x**23 + 4200*c
**(21/2)*d**32*x**21 + 7350*c**(19/2)*d**33*x**19 + 8820*c**(17/2)*d**34*x**17 +
7350*c**(15/2)*d**35*x**15 + 4200*c**(13/2)*d**36*x**13 + 1575*c**(11/2)*d**37*
x**11 + 350*c**(9/2)*d**38*x**9 + 35*c**(7/2)*d**39*x**7))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9), x)