3.1936 \(\int \frac {1}{(a+\frac {b}{x^2})^{3/2} x^9} \, dx\)
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*(a+b/x^2)^(3/2)/b^4-1/5*(a+b/x^2)^(5/2)/b^4-a^3/b^4/(a+b/x^2)^(1/2)-3*a^2*(a+b/x^2)^(1/2)/b^4
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used =
{266, 43} \[ -\frac {a^3}{b^4 \sqrt {a+\frac {b}{x^2}}}-\frac {3 a^2 \sqrt {a+\frac {b}{x^2}}}{b^4}+\frac {a \left (a+\frac {b}{x^2}\right )^{3/2}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4} \]
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)
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{3/2}}+\frac {3 a^2}{b^3 \sqrt {a+b x}}-\frac {3 a \sqrt {a+b x}}{b^3}+\frac {(a+b x)^{3/2}}{b^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\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}\\ \end {align*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.84, size = 63, normalized size = 0.86 \[ -\frac {{\left (16 \, a^{3} x^{6} + 8 \, a^{2} b x^{4} - 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{5 \, {\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{9}}\,{d x} \]
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)
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maple [A] time = 0.01, size = 59, normalized size = 0.81 \[ -\frac {\left (a \,x^{2}+b \right ) \left (16 a^{3} x^{6}+8 a^{2} b \,x^{4}-2 a \,b^{2} x^{2}+b^{3}\right )}{5 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b^{4} x^{8}} \]
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 = 0.87, size = 63, normalized size = 0.86 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, b^{4}} + \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a}{b^{4}} - \frac {3 \, \sqrt {a + \frac {b}{x^{2}}} a^{2}}{b^{4}} - \frac {a^{3}}{\sqrt {a + \frac {b}{x^{2}}} b^{4}} \]
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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mupad [B] time = 1.41, size = 56, normalized size = 0.77 \[ -\frac {\sqrt {a+\frac {b}{x^2}}\,\left (16\,a^3\,x^6+8\,a^2\,b\,x^4-2\,a\,b^2\,x^2+b^3\right )}{5\,b^4\,x^4\,\left (a\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^9*(a + b/x^2)^(3/2)),x)
[Out]
-((a + b/x^2)^(1/2)*(b^3 + 16*a^3*x^6 - 2*a*b^2*x^2 + 8*a^2*b*x^4))/(5*b^4*x^4*(b + a*x^2))
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sympy [B] time = 4.25, 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**1
7 + 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 + 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) - 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*s
qrt(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**(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) + 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**1
1 + 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 + 75*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**(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) + 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 + 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) +
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**(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)
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