3.1921 \(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^9} \, dx\)
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^2*(a+b/x^2)^(3/2)/b^4+3/5*a*(a+b/x^2)^(5/2)/b^4-1/7*(a+b/x^2)^(7/2)/b^4+a^3*(a+b/x^2)^(1/2)/b^4
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 75, 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} \[ -\frac {a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}{b^4}+\frac {a^3 \sqrt {a+\frac {b}{x^2}}}{b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{7/2}}{7 b^4}+\frac {3 a \left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4} \]
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)
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}{\sqrt {a+\frac {b}{x^2}} x^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 \sqrt {a+b x}}+\frac {3 a^2 \sqrt {a+b x}}{b^3}-\frac {3 a (a+b x)^{3/2}}{b^3}+\frac {(a+b x)^{5/2}}{b^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\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 {3 a \left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^4}-\frac {\left (a+\frac {b}{x^2}\right )^{7/2}}{7 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, 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)
________________________________________________________________________________________
fricas [A] time = 1.42, size = 53, normalized size = 0.71 \[ \frac {{\left (16 \, a^{3} x^{6} - 8 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 5 \, b^{3}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{35 \, b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^2)^(1/2)/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)
________________________________________________________________________________________
giac [A] time = 0.23, size = 123, normalized size = 1.64 \[ \frac {70 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )}^{3} a^{\frac {3}{2}} + 84 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )}^{2} a b + 35 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} b^{2} + 5 \, b^{3}}{35 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^2)^(1/2)/x^9,x, algorithm="giac")
[Out]
1/35*(70*(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))^3*a^(3/2) + 84*(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))^2*a*b + 35*(sq
rt(a)*x^2 - sqrt(a*x^4 + b*x^2))*sqrt(a)*b^2 + 5*b^3)/(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))^7
________________________________________________________________________________________
maple [A] time = 0.01, size = 61, normalized size = 0.81 \[ \frac {\left (a \,x^{2}+b \right ) \left (16 a^{3} x^{6}-8 a^{2} b \,x^{4}+6 a \,b^{2} x^{2}-5 b^{3}\right )}{35 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, b^{4} x^{8}} \]
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)
________________________________________________________________________________________
maxima [A] time = 0.90, size = 63, normalized size = 0.84 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}}}{7 \, b^{4}} + \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a}{5 \, b^{4}} - \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2}}{b^{4}} + \frac {\sqrt {a + \frac {b}{x^{2}}} a^{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^2)^(1/2)/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
________________________________________________________________________________________
mupad [B] time = 1.32, size = 73, normalized size = 0.97 \[ \frac {16\,a^3\,\sqrt {a+\frac {b}{x^2}}}{35\,b^4}-\frac {\sqrt {a+\frac {b}{x^2}}}{7\,b\,x^6}+\frac {6\,a\,\sqrt {a+\frac {b}{x^2}}}{35\,b^2\,x^4}-\frac {8\,a^2\,\sqrt {a+\frac {b}{x^2}}}{35\,b^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^9*(a + b/x^2)^(1/2)),x)
[Out]
(16*a^3*(a + b/x^2)^(1/2))/(35*b^4) - (a + b/x^2)^(1/2)/(7*b*x^6) + (6*a*(a + b/x^2)^(1/2))/(35*b^2*x^4) - (8*
a^2*(a + b/x^2)^(1/2))/(35*b^3*x^2)
________________________________________________________________________________________
sympy [B] time = 3.56, 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 + 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) + 140*a**(17/2)*b**(31/2)*x**1
0*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) + 21*a**(
15/2)*b**(33/2)*x**8*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) - 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 + 21
0*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 + 525*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)/(3
5*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**(1
9/2)*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**17*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)
________________________________________________________________________________________