3.1934 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^9} \, dx\)

Optimal. Leaf size=73 \[ -\frac{a^3}{b^4 \sqrt{a+\frac{b}{x^2}}}-\frac{3 a^2 \sqrt{a+\frac{b}{x^2}}}{b^4}+\frac{a \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]

[Out]

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

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Rubi [A]  time = 0.115064, antiderivative size = 73, 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 \[ -\frac{a^3}{b^4 \sqrt{a+\frac{b}{x^2}}}-\frac{3 a^2 \sqrt{a+\frac{b}{x^2}}}{b^4}+\frac{a \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^9),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.0021, size = 65, normalized size = 0.89 \[ - \frac{a^{3}}{b^{4} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{b^{4}} + \frac{a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{b^{4}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.0402443, size = 53, normalized size = 0.73 \[ \frac{-16 a^3 x^6-8 a^2 b x^4+2 a b^2 x^2-b^3}{5 b^4 x^6 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^9),x]

[Out]

(-b^3 + 2*a*b^2*x^2 - 8*a^2*b*x^4 - 16*a^3*x^6)/(5*b^4*Sqrt[a + b/x^2]*x^6)

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Maple [A]  time = 0.009, size = 59, normalized size = 0.8 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 16\,{a}^{3}{x}^{6}+8\,{a}^{2}b{x}^{4}-2\,a{b}^{2}{x}^{2}+{b}^{3} \right ) }{5\,{x}^{8}{b}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^9,x)

[Out]

-1/5*(a*x^2+b)*(16*a^3*x^6+8*a^2*b*x^4-2*a*b^2*x^2+b^3)/x^8/b^4/((a*x^2+b)/x^2)^
(3/2)

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Maxima [A]  time = 1.42275, size = 85, normalized size = 1.16 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b^{4}} + \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a}{b^{4}} - \frac{3 \, \sqrt{a + \frac{b}{x^{2}}} a^{2}}{b^{4}} - \frac{a^{3}}{\sqrt{a + \frac{b}{x^{2}}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^9),x, algorithm="maxima")

[Out]

-1/5*(a + b/x^2)^(5/2)/b^4 + (a + b/x^2)^(3/2)*a/b^4 - 3*sqrt(a + b/x^2)*a^2/b^4
- a^3/(sqrt(a + b/x^2)*b^4)

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Fricas [A]  time = 0.243883, size = 85, normalized size = 1.16 \[ -\frac{{\left (16 \, a^{3} x^{6} + 8 \, a^{2} b x^{4} - 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \,{\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^9),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.3459, size = 1844, normalized size = 25.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**9,x)

[Out]

-16*a**(21/2)*b**(23/2)*x**16*sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a
**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75
*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) - 88*a**(
19/2)*b**(25/2)*x**14*sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)
*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2
)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) - 198*a**(17/2)*b
**(27/2)*x**12*sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*
x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19
*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) - 231*a**(15/2)*b**(29/2
)*x**10*sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 +
75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 +
30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) - 145*a**(13/2)*b**(31/2)*x**8*
sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(
13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(
7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) - 46*a**(11/2)*b**(33/2)*x**6*sqrt(a*x*
*2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**
17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**2
0*x**7 + 5*a**(5/2)*b**21*x**5) - 8*a**(9/2)*b**(35/2)*x**4*sqrt(a*x**2/b + 1)/(
5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 +
100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*
a**(5/2)*b**21*x**5) - 3*a**(7/2)*b**(37/2)*x**2*sqrt(a*x**2/b + 1)/(5*a**(17/2)
*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/
2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b*
*21*x**5) - a**(5/2)*b**(39/2)*sqrt(a*x**2/b + 1)/(5*a**(17/2)*b**15*x**17 + 30*
a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 7
5*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) + 16*a**
11*b**11*x**17/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2
)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)
*b**20*x**7 + 5*a**(5/2)*b**21*x**5) + 96*a**10*b**12*x**15/(5*a**(17/2)*b**15*x
**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18
*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5
) + 240*a**9*b**13*x**13/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 7
5*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 3
0*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) + 320*a**8*b**14*x**11/(5*a**(17/
2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(1
1/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*
b**21*x**5) + 240*a**7*b**15*x**9/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*
x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19
*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5) + 96*a**6*b**16*x**7/(5*
a**(17/2)*b**15*x**17 + 30*a**(15/2)*b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 10
0*a**(11/2)*b**18*x**11 + 75*a**(9/2)*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a*
*(5/2)*b**21*x**5) + 16*a**5*b**17*x**5/(5*a**(17/2)*b**15*x**17 + 30*a**(15/2)*
b**16*x**15 + 75*a**(13/2)*b**17*x**13 + 100*a**(11/2)*b**18*x**11 + 75*a**(9/2)
*b**19*x**9 + 30*a**(7/2)*b**20*x**7 + 5*a**(5/2)*b**21*x**5)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^9),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^9), x)