∖int∖left(a + b _ x2 ∖right)∖left(c + d

3.784 \(\int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x^{12} \, dx\)

Optimal. Leaf size=150 \[ -\frac{16 d^3 x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{15015 c^5}+\frac{8 d^2 x^7 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{3003 c^4}-\frac{2 d x^9 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{429 c^3}+\frac{x^{11} \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{143 c^2}+\frac{a x^{13} \left (c+\frac{d}{x^2}\right )^{5/2}}{13 c} \]

[Out]

(-16*d^3*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^5)/(15015*c^5) + (8*d^2*(13*b*c -

8*a*d)*(c + d/x^2)^(5/2)*x^7)/(3003*c^4) - (2*d*(13*b*c - 8*a*d)*(c + d/x^2)^(5/

2)*x^9)/(429*c^3) + ((13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^11)/(143*c^2) + (a*(c

+ d/x^2)^(5/2)*x^13)/(13*c)

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Rubi [A] time = 0.262444, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{16 d^3 x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{15015 c^5}+\frac{8 d^2 x^7 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{3003 c^4}-\frac{2 d x^9 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{429 c^3}+\frac{x^{11} \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{143 c^2}+\frac{a x^{13} \left (c+\frac{d}{x^2}\right )^{5/2}}{13 c} \]

Antiderivative was successfully veriﬁed.

[In] Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]

[Out]

(-16*d^3*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^5)/(15015*c^5) + (8*d^2*(13*b*c -

8*a*d)*(c + d/x^2)^(5/2)*x^7)/(3003*c^4) - (2*d*(13*b*c - 8*a*d)*(c + d/x^2)^(5/

2)*x^9)/(429*c^3) + ((13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^11)/(143*c^2) + (a*(c

+ d/x^2)^(5/2)*x^13)/(13*c)

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Rubi in Sympy [A] time = 20.4799, size = 146, normalized size = 0.97 \[ \frac{a x^{13} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{13 c} - \frac{x^{11} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (8 a d - 13 b c\right )}{143 c^{2}} + \frac{2 d x^{9} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (8 a d - 13 b c\right )}{429 c^{3}} - \frac{8 d^{2} x^{7} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (8 a d - 13 b c\right )}{3003 c^{4}} + \frac{16 d^{3} x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (8 a d - 13 b c\right )}{15015 c^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**12,x)

[Out]

a*x**13*(c + d/x**2)**(5/2)/(13*c) - x**11*(c + d/x**2)**(5/2)*(8*a*d - 13*b*c)/

(143*c**2) + 2*d*x**9*(c + d/x**2)**(5/2)*(8*a*d - 13*b*c)/(429*c**3) - 8*d**2*x

**7*(c + d/x**2)**(5/2)*(8*a*d - 13*b*c)/(3003*c**4) + 16*d**3*x**5*(c + d/x**2)

**(5/2)*(8*a*d - 13*b*c)/(15015*c**5)

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Mathematica [A] time = 0.0966569, size = 110, normalized size = 0.73 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (a \left (1155 c^4 x^8-840 c^3 d x^6+560 c^2 d^2 x^4-320 c d^3 x^2+128 d^4\right )+13 b c \left (105 c^3 x^6-70 c^2 d x^4+40 c d^2 x^2-16 d^3\right )\right )}{15015 c^5} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(13*b*c*(-16*d^3 + 40*c*d^2*x^2 - 70*c^2*d*x^4

+ 105*c^3*x^6) + a*(128*d^4 - 320*c*d^3*x^2 + 560*c^2*d^2*x^4 - 840*c^3*d*x^6 +

1155*c^4*x^8)))/(15015*c^5)

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Maple [A] time = 0.011, size = 115, normalized size = 0.8 \[{\frac{{x}^{3} \left ( 1155\,a{x}^{8}{c}^{4}-840\,a{c}^{3}d{x}^{6}+1365\,b{c}^{4}{x}^{6}+560\,a{c}^{2}{d}^{2}{x}^{4}-910\,b{c}^{3}d{x}^{4}-320\,ac{d}^{3}{x}^{2}+520\,b{c}^{2}{d}^{2}{x}^{2}+128\,a{d}^{4}-208\,bc{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{15015\,{c}^{5}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x)

[Out]

1/15015*((c*x^2+d)/x^2)^(3/2)*x^3*(1155*a*c^4*x^8-840*a*c^3*d*x^6+1365*b*c^4*x^6

+560*a*c^2*d^2*x^4-910*b*c^3*d*x^4-320*a*c*d^3*x^2+520*b*c^2*d^2*x^2+128*a*d^4-2

08*b*c*d^3)*(c*x^2+d)/c^5

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Maxima [A] time = 1.38693, size = 213, normalized size = 1.42 \[ \frac{{\left (105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}} x^{11} - 385 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} d x^{9} + 495 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d^{2} x^{7} - 231 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{3} x^{5}\right )} b}{1155 \, c^{4}} + \frac{{\left (1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{13}{2}} x^{13} - 5460 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}} d x^{11} + 10010 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} d^{2} x^{9} - 8580 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d^{3} x^{7} + 3003 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{4} x^{5}\right )} a}{15015 \, c^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x, algorithm="maxima")

[Out]

1/1155*(105*(c + d/x^2)^(11/2)*x^11 - 385*(c + d/x^2)^(9/2)*d*x^9 + 495*(c + d/x

^2)^(7/2)*d^2*x^7 - 231*(c + d/x^2)^(5/2)*d^3*x^5)*b/c^4 + 1/15015*(1155*(c + d/

x^2)^(13/2)*x^13 - 5460*(c + d/x^2)^(11/2)*d*x^11 + 10010*(c + d/x^2)^(9/2)*d^2*

x^9 - 8580*(c + d/x^2)^(7/2)*d^3*x^7 + 3003*(c + d/x^2)^(5/2)*d^4*x^5)*a/c^5

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Fricas [A] time = 0.227339, size = 209, normalized size = 1.39 \[ \frac{{\left (1155 \, a c^{6} x^{13} + 105 \,{\left (13 \, b c^{6} + 14 \, a c^{5} d\right )} x^{11} + 35 \,{\left (52 \, b c^{5} d + a c^{4} d^{2}\right )} x^{9} + 5 \,{\left (13 \, b c^{4} d^{2} - 8 \, a c^{3} d^{3}\right )} x^{7} - 6 \,{\left (13 \, b c^{3} d^{3} - 8 \, a c^{2} d^{4}\right )} x^{5} + 8 \,{\left (13 \, b c^{2} d^{4} - 8 \, a c d^{5}\right )} x^{3} - 16 \,{\left (13 \, b c d^{5} - 8 \, a d^{6}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15015 \, c^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x, algorithm="fricas")

[Out]

1/15015*(1155*a*c^6*x^13 + 105*(13*b*c^6 + 14*a*c^5*d)*x^11 + 35*(52*b*c^5*d + a

*c^4*d^2)*x^9 + 5*(13*b*c^4*d^2 - 8*a*c^3*d^3)*x^7 - 6*(13*b*c^3*d^3 - 8*a*c^2*d

^4)*x^5 + 8*(13*b*c^2*d^4 - 8*a*c*d^5)*x^3 - 16*(13*b*c*d^5 - 8*a*d^6)*x)*sqrt((

c*x^2 + d)/x^2)/c^5

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Sympy [A] time = 35.9518, size = 3351, normalized size = 22.34 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**12,x)

[Out]

693*a*c**12*d**(51/2)*x**22*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c

**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**

29*x**2 + 9009*c**6*d**30) + 3528*a*c**11*d**(53/2)*x**20*sqrt(c*x**2/d + 1)/(90

09*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c*

*8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 7175*a*c**10*d**(55/2

)*x**18*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90

090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*

d**30) + 7290*a*c**9*d**(57/2)*x**16*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10

+ 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045

*c**7*d**29*x**2 + 9009*c**6*d**30) + 315*a*c**9*d**(35/2)*x**18*sqrt(c*x**2/d +

1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 1386

0*c**6*d**19*x**2 + 3465*c**5*d**20) + 3699*a*c**8*d**(59/2)*x**14*sqrt(c*x**2/d

+ 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 +

90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 1295*a*c**8*

d**(37/2)*x**16*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6

+ 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 756*a*c**7

*d**(61/2)*x**12*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*

x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9

009*c**6*d**30) + 1990*a*c**7*d**(39/2)*x**14*sqrt(c*x**2/d + 1)/(3465*c**9*d**1

6*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 +

3465*c**5*d**20) - 63*a*c**6*d**(63/2)*x**10*sqrt(c*x**2/d + 1)/(9009*c**11*d**

25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**

4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 1358*a*c**6*d**(41/2)*x**12*sqrt(

c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x

**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) - 630*a*c**5*d**(65/2)*x**8*sqrt(

c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**2

7*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 343*

a*c**5*d**(43/2)*x**10*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**

17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) - 168

0*a*c**4*d**(67/2)*x**6*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10

*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x

**2 + 9009*c**6*d**30) + 35*a*c**4*d**(45/2)*x**8*sqrt(c*x**2/d + 1)/(3465*c**9*

d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x*

*2 + 3465*c**5*d**20) - 2016*a*c**3*d**(69/2)*x**4*sqrt(c*x**2/d + 1)/(9009*c**1

1*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**2

8*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 280*a*c**3*d**(47/2)*x**6*sq

rt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**1

8*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) - 1152*a*c**2*d**(71/2)*x**2*s

qrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*

d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) +

560*a*c**2*d**(49/2)*x**4*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*

d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) -

256*a*c*d**(73/2)*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26

*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 +

9009*c**6*d**30) + 448*a*c*d**(51/2)*x**2*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x*

*8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 346

5*c**5*d**20) + 128*a*d**(53/2)*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860

*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**

20) + 315*b*c**10*d**(33/2)*x**18*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 138

60*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d

**20) + 1295*b*c**9*d**(35/2)*x**16*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1

3860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5

*d**20) + 1990*b*c**8*d**(37/2)*x**14*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 +

13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c*

*5*d**20) + 1358*b*c**7*d**(39/2)*x**12*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8

+ 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*

c**5*d**20) + 35*b*c**7*d**(21/2)*x**14*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 +

945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 343*b*c**6*d**(41

/2)*x**10*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 207

90*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 110*b*c**6*d**(2

3/2)*x**12*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c*

*5*d**11*x**2 + 315*c**4*d**12) + 35*b*c**5*d**(43/2)*x**8*sqrt(c*x**2/d + 1)/(3

465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6

*d**19*x**2 + 3465*c**5*d**20) + 114*b*c**5*d**(25/2)*x**10*sqrt(c*x**2/d + 1)/(

315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)

+ 280*b*c**4*d**(45/2)*x**6*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c*

*8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20)

+ 40*b*c**4*d**(27/2)*x**8*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d*

*10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 560*b*c**3*d**(47/2)*x**4*sqr

t(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18

*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) - 5*b*c**3*d**(29/2)*x**6*sqrt(

c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 +

315*c**4*d**12) + 448*b*c**2*d**(49/2)*x**2*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*

x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3

465*c**5*d**20) - 30*b*c**2*d**(31/2)*x**4*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**

6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 128*b*c*d**(51

/2)*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**

7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) - 40*b*c*d**(33/2)*x**2*

sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x*

*2 + 315*c**4*d**12) - 16*b*d**(35/2)*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 9

45*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)

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GIAC/XCAS [A] time = 0.237939, size = 440, normalized size = 2.93 \[ \frac{\frac{143 \,{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} b d{\rm sign}\left (x\right )}{c^{3}} + \frac{13 \,{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} b{\rm sign}\left (x\right )}{c^{3}} + \frac{13 \,{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} a d{\rm sign}\left (x\right )}{c^{4}} + \frac{5 \,{\left (693 \,{\left (c x^{2} + d\right )}^{\frac{13}{2}} - 4095 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{5}\right )} a{\rm sign}\left (x\right )}{c^{4}}}{45045 \, c} + \frac{16 \,{\left (13 \, b c d^{\frac{11}{2}} - 8 \, a d^{\frac{13}{2}}\right )}{\rm sign}\left (x\right )}{15015 \, c^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x, algorithm="giac")

[Out]

1/45045*(143*(35*(c*x^2 + d)^(9/2) - 135*(c*x^2 + d)^(7/2)*d + 189*(c*x^2 + d)^(

5/2)*d^2 - 105*(c*x^2 + d)^(3/2)*d^3)*b*d*sign(x)/c^3 + 13*(315*(c*x^2 + d)^(11/

2) - 1540*(c*x^2 + d)^(9/2)*d + 2970*(c*x^2 + d)^(7/2)*d^2 - 2772*(c*x^2 + d)^(5

/2)*d^3 + 1155*(c*x^2 + d)^(3/2)*d^4)*b*sign(x)/c^3 + 13*(315*(c*x^2 + d)^(11/2)

- 1540*(c*x^2 + d)^(9/2)*d + 2970*(c*x^2 + d)^(7/2)*d^2 - 2772*(c*x^2 + d)^(5/2

)*d^3 + 1155*(c*x^2 + d)^(3/2)*d^4)*a*d*sign(x)/c^4 + 5*(693*(c*x^2 + d)^(13/2)

- 4095*(c*x^2 + d)^(11/2)*d + 10010*(c*x^2 + d)^(9/2)*d^2 - 12870*(c*x^2 + d)^(7

/2)*d^3 + 9009*(c*x^2 + d)^(5/2)*d^4 - 3003*(c*x^2 + d)^(3/2)*d^5)*a*sign(x)/c^4

)/c + 16/15015*(13*b*c*d^(11/2) - 8*a*d^(13/2))*sign(x)/c^5

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