3.768 \(\int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^{10} \, dx\)

Optimal. Leaf size=150 \[ -\frac{16 d^3 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac{8 d^2 x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac{2 d x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac{x^9 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}+\frac{a x^{11} \left (c+\frac{d}{x^2}\right )^{3/2}}{11 c} \]

[Out]

(-16*d^3*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^3)/(3465*c^5) + (8*d^2*(11*b*c - 8
*a*d)*(c + d/x^2)^(3/2)*x^5)/(1155*c^4) - (2*d*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2
)*x^7)/(231*c^3) + ((11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^9)/(99*c^2) + (a*(c + d
/x^2)^(3/2)*x^11)/(11*c)

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Rubi [A]  time = 0.256791, 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 \[ -\frac{16 d^3 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac{8 d^2 x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac{2 d x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac{x^9 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}+\frac{a x^{11} \left (c+\frac{d}{x^2}\right )^{3/2}}{11 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^10,x]

[Out]

(-16*d^3*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^3)/(3465*c^5) + (8*d^2*(11*b*c - 8
*a*d)*(c + d/x^2)^(3/2)*x^5)/(1155*c^4) - (2*d*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2
)*x^7)/(231*c^3) + ((11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^9)/(99*c^2) + (a*(c + d
/x^2)^(3/2)*x^11)/(11*c)

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Rubi in Sympy [A]  time = 18.0129, size = 146, normalized size = 0.97 \[ \frac{a x^{11} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{11 c} - \frac{x^{9} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (8 a d - 11 b c\right )}{99 c^{2}} + \frac{2 d x^{7} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (8 a d - 11 b c\right )}{231 c^{3}} - \frac{8 d^{2} x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (8 a d - 11 b c\right )}{1155 c^{4}} + \frac{16 d^{3} x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (8 a d - 11 b c\right )}{3465 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x**10*(c+d/x**2)**(1/2),x)

[Out]

a*x**11*(c + d/x**2)**(3/2)/(11*c) - x**9*(c + d/x**2)**(3/2)*(8*a*d - 11*b*c)/(
99*c**2) + 2*d*x**7*(c + d/x**2)**(3/2)*(8*a*d - 11*b*c)/(231*c**3) - 8*d**2*x**
5*(c + d/x**2)**(3/2)*(8*a*d - 11*b*c)/(1155*c**4) + 16*d**3*x**3*(c + d/x**2)**
(3/2)*(8*a*d - 11*b*c)/(3465*c**5)

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Mathematica [A]  time = 0.0987397, size = 108, normalized size = 0.72 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (a \left (315 c^4 x^8-280 c^3 d x^6+240 c^2 d^2 x^4-192 c d^3 x^2+128 d^4\right )+11 b c \left (35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right )\right )}{3465 c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^10,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)*(11*b*c*(-16*d^3 + 24*c*d^2*x^2 - 30*c^2*d*x^4 +
35*c^3*x^6) + a*(128*d^4 - 192*c*d^3*x^2 + 240*c^2*d^2*x^4 - 280*c^3*d*x^6 + 315
*c^4*x^8)))/(3465*c^5)

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Maple [A]  time = 0.011, size = 113, normalized size = 0.8 \[{\frac{x \left ( 315\,a{x}^{8}{c}^{4}-280\,a{c}^{3}d{x}^{6}+385\,b{c}^{4}{x}^{6}+240\,a{c}^{2}{d}^{2}{x}^{4}-330\,b{c}^{3}d{x}^{4}-192\,ac{d}^{3}{x}^{2}+264\,b{c}^{2}{d}^{2}{x}^{2}+128\,a{d}^{4}-176\,bc{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{c}^{5}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^10*(c+d/x^2)^(1/2),x)

[Out]

1/3465*((c*x^2+d)/x^2)^(1/2)*x*(315*a*c^4*x^8-280*a*c^3*d*x^6+385*b*c^4*x^6+240*
a*c^2*d^2*x^4-330*b*c^3*d*x^4-192*a*c*d^3*x^2+264*b*c^2*d^2*x^2+128*a*d^4-176*b*
c*d^3)*(c*x^2+d)/c^5

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Maxima [A]  time = 1.37017, size = 213, normalized size = 1.42 \[ \frac{{\left (35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} x^{9} - 135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d x^{7} + 189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{2} x^{5} - 105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{3} x^{3}\right )} b}{315 \, c^{4}} + \frac{{\left (315 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}} x^{11} - 1540 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} d x^{9} + 2970 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d^{2} x^{7} - 2772 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{3} x^{5} + 1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{4} x^{3}\right )} a}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^10,x, algorithm="maxima")

[Out]

1/315*(35*(c + d/x^2)^(9/2)*x^9 - 135*(c + d/x^2)^(7/2)*d*x^7 + 189*(c + d/x^2)^
(5/2)*d^2*x^5 - 105*(c + d/x^2)^(3/2)*d^3*x^3)*b/c^4 + 1/3465*(315*(c + d/x^2)^(
11/2)*x^11 - 1540*(c + d/x^2)^(9/2)*d*x^9 + 2970*(c + d/x^2)^(7/2)*d^2*x^7 - 277
2*(c + d/x^2)^(5/2)*d^3*x^5 + 1155*(c + d/x^2)^(3/2)*d^4*x^3)*a/c^5

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Fricas [A]  time = 0.230501, size = 177, normalized size = 1.18 \[ \frac{{\left (315 \, a c^{5} x^{11} + 35 \,{\left (11 \, b c^{5} + a c^{4} d\right )} x^{9} + 5 \,{\left (11 \, b c^{4} d - 8 \, a c^{3} d^{2}\right )} x^{7} - 6 \,{\left (11 \, b c^{3} d^{2} - 8 \, a c^{2} d^{3}\right )} x^{5} + 8 \,{\left (11 \, b c^{2} d^{3} - 8 \, a c d^{4}\right )} x^{3} - 16 \,{\left (11 \, b c d^{4} - 8 \, a d^{5}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^10,x, algorithm="fricas")

[Out]

1/3465*(315*a*c^5*x^11 + 35*(11*b*c^5 + a*c^4*d)*x^9 + 5*(11*b*c^4*d - 8*a*c^3*d
^2)*x^7 - 6*(11*b*c^3*d^2 - 8*a*c^2*d^3)*x^5 + 8*(11*b*c^2*d^3 - 8*a*c*d^4)*x^3
- 16*(11*b*c*d^4 - 8*a*d^5)*x)*sqrt((c*x^2 + d)/x^2)/c^5

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Sympy [A]  time = 15.5781, size = 1386, normalized size = 9.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**10*(c+d/x**2)**(1/2),x)

[Out]

315*a*c**9*d**(33/2)*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 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) +
1295*a*c**8*d**(35/2)*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)
+ 1990*a*c**7*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**2
0) + 1358*a*c**6*d**(39/2)*x**12*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1386
0*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) + 343*a*c**5*d**(41/2)*x**10*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) + 35*a*c**4*d**(43/2)*x**8*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1386
0*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) + 280*a*c**3*d**(45/2)*x**6*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1386
0*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) + 560*a*c**2*d**(47/2)*x**4*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1386
0*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) + 448*a*c*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 + 3465*c**5*d**20
) + 128*a*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) + 35*b*c
**7*d**(19/2)*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) + 110*b*c**6*d**(21/2)*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) + 114*b*c**5*d**(23/2)*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) + 40*b*c**4*d**(25/
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) - 5*b*c**3*d**(27/2)*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) - 30
*b*c**2*d**(29/2)*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) - 40*b*c*d**(31/2)*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**(33/2)*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**1
0*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)

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GIAC/XCAS [A]  time = 0.230706, size = 217, normalized size = 1.45 \[ \frac{\frac{11 \,{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} b{\rm sign}\left (x\right )}{c^{3}} + \frac{{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} a{\rm sign}\left (x\right )}{c^{4}}}{3465 \, c} + \frac{16 \,{\left (11 \, b c d^{\frac{9}{2}} - 8 \, a d^{\frac{11}{2}}\right )}{\rm sign}\left (x\right )}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^10,x, algorithm="giac")

[Out]

1/3465*(11*(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*sign(x)/c^3 + (315*(c*x^2 + d)^(11/2) - 15
40*(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*sign(x)/c^4)/c + 16/3465*(11*b*c*d^(9/2) - 8*a*
d^(11/2))*sign(x)/c^5