3.1919 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^9} \, dx\)
Optimal. Leaf size=75 \[ \frac{a^3 \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b^4}+\frac{3 a \left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]
[Out]
(a^3*Sqrt[a + b/x^2])/b^4 - (a^2*(a + b/x^2)^(3/2))/b^4 + (3*a*(a + b/x^2)^(5/2)
)/(5*b^4) - (a + b/x^2)^(7/2)/(7*b^4)
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Rubi [A] time = 0.116408, antiderivative size = 75, 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
\[ \frac{a^3 \sqrt{a+\frac{b}{x^2}}}{b^4}-\frac{a^2 \left (a+\frac{b}{x^2}\right )^{3/2}}{b^4}-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b^4}+\frac{3 a \left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^2]*x^9),x]
[Out]
(a^3*Sqrt[a + b/x^2])/b^4 - (a^2*(a + b/x^2)^(3/2))/b^4 + (3*a*(a + b/x^2)^(5/2)
)/(5*b^4) - (a + b/x^2)^(7/2)/(7*b^4)
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Rubi in Sympy [A] time = 14.1043, size = 66, normalized size = 0.88 \[ \frac{a^{3} \sqrt{a + \frac{b}{x^{2}}}}{b^{4}} - \frac{a^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{b^{4}} + \frac{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{7}{2}}}{7 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(1/2)/x**9,x)
[Out]
a**3*sqrt(a + b/x**2)/b**4 - a**2*(a + b/x**2)**(3/2)/b**4 + 3*a*(a + b/x**2)**(
5/2)/(5*b**4) - (a + b/x**2)**(7/2)/(7*b**4)
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Mathematica [A] time = 0.046307, size = 53, normalized size = 0.71 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (16 a^3 x^6-8 a^2 b x^4+6 a b^2 x^2-5 b^3\right )}{35 b^4 x^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^2]*x^9),x]
[Out]
(Sqrt[a + b/x^2]*(-5*b^3 + 6*a*b^2*x^2 - 8*a^2*b*x^4 + 16*a^3*x^6))/(35*b^4*x^6)
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Maple [A] time = 0.009, size = 61, normalized size = 0.8 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 16\,{a}^{3}{x}^{6}-8\,{a}^{2}b{x}^{4}+6\,a{b}^{2}{x}^{2}-5\,{b}^{3} \right ) }{35\,{x}^{8}{b}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(1/2)/x^9,x)
[Out]
1/35*(a*x^2+b)*(16*a^3*x^6-8*a^2*b*x^4+6*a*b^2*x^2-5*b^3)/x^8/b^4/((a*x^2+b)/x^2
)^(1/2)
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Maxima [A] time = 1.43946, size = 85, normalized size = 1.13 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{7}{2}}}{7 \, b^{4}} + \frac{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} a}{5 \, b^{4}} - \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{2}}{b^{4}} + \frac{\sqrt{a + \frac{b}{x^{2}}} a^{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^9),x, algorithm="maxima")
[Out]
-1/7*(a + b/x^2)^(7/2)/b^4 + 3/5*(a + b/x^2)^(5/2)*a/b^4 - (a + b/x^2)^(3/2)*a^2
/b^4 + sqrt(a + b/x^2)*a^3/b^4
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Fricas [A] time = 0.246694, size = 72, normalized size = 0.96 \[ \frac{{\left (16 \, a^{3} x^{6} - 8 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 5 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{35 \, b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^9),x, algorithm="fricas")
[Out]
1/35*(16*a^3*x^6 - 8*a^2*b*x^4 + 6*a*b^2*x^2 - 5*b^3)*sqrt((a*x^2 + b)/x^2)/(b^4
*x^6)
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Sympy [A] time = 15.298, size = 1969, normalized size = 26.25 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(1/2)/x**9,x)
[Out]
16*a**(25/2)*b**(23/2)*x**18*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*
a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) +
88*a**(23/2)*b**(25/2)*x**16*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*
a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) +
198*a**(21/2)*b**(27/2)*x**14*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210
*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) +
231*a**(19/2)*b**(29/2)*x**12*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 21
0*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13
+ 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7)
+ 140*a**(17/2)*b**(31/2)*x**10*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 2
10*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13
+ 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7)
+ 21*a**(15/2)*b**(33/2)*x**8*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 21
0*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13
+ 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7)
- 42*a**(13/2)*b**(35/2)*x**6*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210
*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) -
47*a**(11/2)*b**(37/2)*x**4*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*
a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 +
525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) -
24*a**(9/2)*b**(39/2)*x**2*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a*
*(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 52
5*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 5*
a**(7/2)*b**(41/2)*sqrt(a*x**2/b + 1)/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*
b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11
/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 16*a**13*b
**11*x**19/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)
*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9
/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 96*a**12*b**12*x**17/(35*a**(19/2)*b*
*15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2
)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2
)*b**21*x**7) - 240*a**11*b**13*x**15/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*
b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11
/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 320*a**10*
b**14*x**13/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2
)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(
9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 240*a**9*b**15*x**11/(35*a**(19/2)*b
**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/
2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/
2)*b**21*x**7) - 96*a**8*b**16*x**9/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b*
*16*x**17 + 525*a**(15/2)*b**17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2
)*b**19*x**11 + 210*a**(9/2)*b**20*x**9 + 35*a**(7/2)*b**21*x**7) - 16*a**7*b**1
7*x**7/(35*a**(19/2)*b**15*x**19 + 210*a**(17/2)*b**16*x**17 + 525*a**(15/2)*b**
17*x**15 + 700*a**(13/2)*b**18*x**13 + 525*a**(11/2)*b**19*x**11 + 210*a**(9/2)*
b**20*x**9 + 35*a**(7/2)*b**21*x**7)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^9),x, algorithm="giac")
[Out]
integrate(1/(sqrt(a + b/x^2)*x^9), x)