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^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)
________________________________________________________________________________________
Rubi [A] time = 0.0383933, 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, 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 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]]
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])
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*}
Mathematica [A] time = 0.0107584, size = 53, normalized size = 0.73 \[ \frac{-8 a^2 b x^4-16 a^3 x^6+2 a b^2 x^2-b^3}{5 b^4 x^6 \sqrt{a+\frac{b}{x^2}}} \]
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)
________________________________________________________________________________________
Maple [A] time = 0.006, size = 59, normalized size = 0.8 \begin{align*} -{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+1/x^2*b)^(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)
________________________________________________________________________________________
Maxima [A] time = 1.03978, size = 85, normalized size = 1.16 \begin{align*} -\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}} \end{align*}
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)
________________________________________________________________________________________
Fricas [A] time = 1.54522, size = 131, normalized size = 1.79 \begin{align*} -\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 )}} \end{align*}
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)
________________________________________________________________________________________
Sympy [B] time = 4.42999, size = 1844, normalized size = 25.26 \begin{align*} \text{result too large to display} \end{align*}
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)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{9}}\,{d x} \end{align*}
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)