3.769 \(\int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^8 \, dx\)

Optimal. Leaf size=117 \[ \frac{8 d^2 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}-\frac{4 d x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac{x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}+\frac{a x^9 \left (c+\frac{d}{x^2}\right )^{3/2}}{9 c} \]

[Out]

(8*d^2*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^3)/(315*c^4) - (4*d*(3*b*c - 2*a*d)*(
c + d/x^2)^(3/2)*x^5)/(105*c^3) + ((3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^7)/(21*c^
2) + (a*(c + d/x^2)^(3/2)*x^9)/(9*c)

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Rubi [A]  time = 0.219225, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{8 d^2 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}-\frac{4 d x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac{x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}+\frac{a x^9 \left (c+\frac{d}{x^2}\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^8,x]

[Out]

(8*d^2*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^3)/(315*c^4) - (4*d*(3*b*c - 2*a*d)*(
c + d/x^2)^(3/2)*x^5)/(105*c^3) + ((3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^7)/(21*c^
2) + (a*(c + d/x^2)^(3/2)*x^9)/(9*c)

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Rubi in Sympy [A]  time = 13.7508, size = 112, normalized size = 0.96 \[ \frac{a x^{9} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{9 c} - \frac{x^{7} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (2 a d - 3 b c\right )}{21 c^{2}} + \frac{4 d x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (2 a d - 3 b c\right )}{105 c^{3}} - \frac{8 d^{2} x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (2 a d - 3 b c\right )}{315 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x**8*(c+d/x**2)**(1/2),x)

[Out]

a*x**9*(c + d/x**2)**(3/2)/(9*c) - x**7*(c + d/x**2)**(3/2)*(2*a*d - 3*b*c)/(21*
c**2) + 4*d*x**5*(c + d/x**2)**(3/2)*(2*a*d - 3*b*c)/(105*c**3) - 8*d**2*x**3*(c
+ d/x**2)**(3/2)*(2*a*d - 3*b*c)/(315*c**4)

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Mathematica [A]  time = 0.083713, size = 86, normalized size = 0.74 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (a \left (35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right )+3 b c \left (15 c^2 x^4-12 c d x^2+8 d^2\right )\right )}{315 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^8,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)*(3*b*c*(8*d^2 - 12*c*d*x^2 + 15*c^2*x^4) + a*(-16
*d^3 + 24*c*d^2*x^2 - 30*c^2*d*x^4 + 35*c^3*x^6)))/(315*c^4)

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Maple [A]  time = 0.011, size = 89, normalized size = 0.8 \[{\frac{x \left ( 35\,a{x}^{6}{c}^{3}-30\,a{c}^{2}d{x}^{4}+45\,b{c}^{3}{x}^{4}+24\,ac{d}^{2}{x}^{2}-36\,b{c}^{2}d{x}^{2}-16\,a{d}^{3}+24\,bc{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{315\,{c}^{4}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^8*(c+d/x^2)^(1/2),x)

[Out]

1/315*((c*x^2+d)/x^2)^(1/2)*x*(35*a*c^3*x^6-30*a*c^2*d*x^4+45*b*c^3*x^4+24*a*c*d
^2*x^2-36*b*c^2*d*x^2-16*a*d^3+24*b*c*d^2)*(c*x^2+d)/c^4

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Maxima [A]  time = 1.39025, size = 167, normalized size = 1.43 \[ \frac{{\left (15 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} x^{7} - 42 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d x^{5} + 35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{2} x^{3}\right )} b}{105 \, c^{3}} + \frac{{\left (35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} x^{9} - 135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d x^{7} + 189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{2} x^{5} - 105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{3} x^{3}\right )} a}{315 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^8,x, algorithm="maxima")

[Out]

1/105*(15*(c + d/x^2)^(7/2)*x^7 - 42*(c + d/x^2)^(5/2)*d*x^5 + 35*(c + d/x^2)^(3
/2)*d^2*x^3)*b/c^3 + 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)*a/c^4

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Fricas [A]  time = 0.233194, size = 144, normalized size = 1.23 \[ \frac{{\left (35 \, a c^{4} x^{9} + 5 \,{\left (9 \, b c^{4} + a c^{3} d\right )} x^{7} + 3 \,{\left (3 \, b c^{3} d - 2 \, a c^{2} d^{2}\right )} x^{5} - 4 \,{\left (3 \, b c^{2} d^{2} - 2 \, a c d^{3}\right )} x^{3} + 8 \,{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^8,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.2474, size = 910, normalized size = 7.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**8*(c+d/x**2)**(1/2),x)

[Out]

35*a*c**7*d**(19/2)*x**14*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) + 110*a*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*a*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*a*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*a*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*a*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*a*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 + 3
15*c**4*d**12) - 16*a*d**(33/2)*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) + 15*b*c**5*d**(9/2)*x**10*
sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 3
3*b*c**4*d**(11/2)*x**8*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x
**2 + 105*c**3*d**6) + 17*b*c**3*d**(13/2)*x**6*sqrt(c*x**2/d + 1)/(105*c**5*d**
4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 3*b*c**2*d**(15/2)*x**4*sqrt(c*x*
*2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 12*b*c*d**
(17/2)*x**2*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c*
*3*d**6) + 8*b*d**(19/2)*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*
x**2 + 105*c**3*d**6)

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GIAC/XCAS [A]  time = 0.214761, size = 180, normalized size = 1.54 \[ \frac{\frac{3 \,{\left (15 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{2}\right )} b{\rm sign}\left (x\right )}{c^{2}} + \frac{{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} a{\rm sign}\left (x\right )}{c^{3}}}{315 \, c} - \frac{8 \,{\left (3 \, b c d^{\frac{7}{2}} - 2 \, a d^{\frac{9}{2}}\right )}{\rm sign}\left (x\right )}{315 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^8,x, algorithm="giac")

[Out]

1/315*(3*(15*(c*x^2 + d)^(7/2) - 42*(c*x^2 + d)^(5/2)*d + 35*(c*x^2 + d)^(3/2)*d
^2)*b*sign(x)/c^2 + (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)*a*sign(x)/c^3)/c - 8/315*(3*b*c*d^(
7/2) - 2*a*d^(9/2))*sign(x)/c^4