\(\int \frac {1}{x^6 (a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [395]

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1}{5 a^4 x^5}+\frac {4 b}{3 a^5 x^3}-\frac {10 b^2}{a^6 x}-\frac {b^3 x}{6 a^4 \left (a+b x^2\right )^3}-\frac {23 b^3 x}{24 a^5 \left (a+b x^2\right )^2}-\frac {71 b^3 x}{16 a^6 \left (a+b x^2\right )}-\frac {231 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}} \] Output:

-1/5/a^4/x^5+4/3*b/a^5/x^3-10*b^2/a^6/x-1/6*b^3*x/a^4/(b*x^2+a)^3-23/24*b^ 
3*x/a^5/(b*x^2+a)^2-71/16*b^3*x/a^6/(b*x^2+a)-231/16*b^(5/2)*arctan(b^(1/2 
)*x/a^(1/2))/a^(13/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {48 a^5-176 a^4 b x^2+1584 a^3 b^2 x^4+7623 a^2 b^3 x^6+9240 a b^4 x^8+3465 b^5 x^{10}}{240 a^6 x^5 \left (a+b x^2\right )^3}-\frac {231 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{13/2}} \] Input:

Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
 

Output:

-1/240*(48*a^5 - 176*a^4*b*x^2 + 1584*a^3*b^2*x^4 + 7623*a^2*b^3*x^6 + 924 
0*a*b^4*x^8 + 3465*b^5*x^10)/(a^6*x^5*(a + b*x^2)^3) - (231*b^(5/2)*ArcTan 
[(Sqrt[b]*x)/Sqrt[a]])/(16*a^(13/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1380, 27, 253, 253, 253, 264, 264, 264, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^4 \int \frac {1}{b^4 x^6 \left (b x^2+a\right )^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^4}dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \int \frac {1}{x^6 \left (b x^2+a\right )^3}dx}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \int \frac {1}{x^6 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{x^6 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\)

Input:

Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
 

Output:

1/(6*a*x^5*(a + b*x^2)^3) + (11*(1/(4*a*x^5*(a + b*x^2)^2) + (9*(1/(2*a*x^ 
5*(a + b*x^2)) + (7*(-1/5*1/(a*x^5) - (b*(-1/3*1/(a*x^3) - (b*(-(1/(a*x)) 
- (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)))/a))/a))/(2*a)))/(4*a)))/ 
(6*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72

method	result	size
default	\(-\frac {b^{3} \left (\frac {\frac {71}{16} x^{5} b^{2}+\frac {59}{6} a b \,x^{3}+\frac {89}{16} a^{2} x}{\left (b \,x^{2}+a \right )^{3}}+\frac {231 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{6}}-\frac {1}{5 a^{4} x^{5}}-\frac {10 b^{2}}{a^{6} x}+\frac {4 b}{3 a^{5} x^{3}}\)	\(87\)
risch	\(\frac {-\frac {231 b^{5} x^{10}}{16 a^{6}}-\frac {77 b^{4} x^{8}}{2 a^{5}}-\frac {2541 b^{3} x^{6}}{80 a^{4}}-\frac {33 b^{2} x^{4}}{5 a^{3}}+\frac {11 b \,x^{2}}{15 a^{2}}-\frac {1}{5 a}}{x^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (b \,x^{2}+a \right )}+\frac {231 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right )}{32 a^{7}}-\frac {231 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right )}{32 a^{7}}\)	\(148\)

Input:

int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/a^6*b^3*((71/16*x^5*b^2+59/6*a*b*x^3+89/16*a^2*x)/(b*x^2+a)^3+231/16/(a 
*b)^(1/2)*arctan(b/(a*b)^(1/2)*x))-1/5/a^4/x^5-10*b^2/a^6/x+4/3*b/a^5/x^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\left [-\frac {6930 \, b^{5} x^{10} + 18480 \, a b^{4} x^{8} + 15246 \, a^{2} b^{3} x^{6} + 3168 \, a^{3} b^{2} x^{4} - 352 \, a^{4} b x^{2} + 96 \, a^{5} - 3465 \, {\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{480 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}, -\frac {3465 \, b^{5} x^{10} + 9240 \, a b^{4} x^{8} + 7623 \, a^{2} b^{3} x^{6} + 1584 \, a^{3} b^{2} x^{4} - 176 \, a^{4} b x^{2} + 48 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 3 \, a b^{4} x^{9} + 3 \, a^{2} b^{3} x^{7} + a^{3} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{240 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}}\right ] \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
 

Output:

[-1/480*(6930*b^5*x^10 + 18480*a*b^4*x^8 + 15246*a^2*b^3*x^6 + 3168*a^3*b^ 
2*x^4 - 352*a^4*b*x^2 + 96*a^5 - 3465*(b^5*x^11 + 3*a*b^4*x^9 + 3*a^2*b^3* 
x^7 + a^3*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + 
a)))/(a^6*b^3*x^11 + 3*a^7*b^2*x^9 + 3*a^8*b*x^7 + a^9*x^5), -1/240*(3465* 
b^5*x^10 + 9240*a*b^4*x^8 + 7623*a^2*b^3*x^6 + 1584*a^3*b^2*x^4 - 176*a^4* 
b*x^2 + 48*a^5 + 3465*(b^5*x^11 + 3*a*b^4*x^9 + 3*a^2*b^3*x^7 + a^3*b^2*x^ 
5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b^3*x^11 + 3*a^7*b^2*x^9 + 3*a^8*b* 
x^7 + a^9*x^5)]
 

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {231 \sqrt {- \frac {b^{5}}{a^{13}}} \log {\left (- \frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} - \frac {231 \sqrt {- \frac {b^{5}}{a^{13}}} \log {\left (\frac {a^{7} \sqrt {- \frac {b^{5}}{a^{13}}}}{b^{3}} + x \right )}}{32} + \frac {- 48 a^{5} + 176 a^{4} b x^{2} - 1584 a^{3} b^{2} x^{4} - 7623 a^{2} b^{3} x^{6} - 9240 a b^{4} x^{8} - 3465 b^{5} x^{10}}{240 a^{9} x^{5} + 720 a^{8} b x^{7} + 720 a^{7} b^{2} x^{9} + 240 a^{6} b^{3} x^{11}} \] Input:

integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
 

Output:

231*sqrt(-b**5/a**13)*log(-a**7*sqrt(-b**5/a**13)/b**3 + x)/32 - 231*sqrt( 
-b**5/a**13)*log(a**7*sqrt(-b**5/a**13)/b**3 + x)/32 + (-48*a**5 + 176*a** 
4*b*x**2 - 1584*a**3*b**2*x**4 - 7623*a**2*b**3*x**6 - 9240*a*b**4*x**8 - 
3465*b**5*x**10)/(240*a**9*x**5 + 720*a**8*b*x**7 + 720*a**7*b**2*x**9 + 2 
40*a**6*b**3*x**11)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {3465 \, b^{5} x^{10} + 9240 \, a b^{4} x^{8} + 7623 \, a^{2} b^{3} x^{6} + 1584 \, a^{3} b^{2} x^{4} - 176 \, a^{4} b x^{2} + 48 \, a^{5}}{240 \, {\left (a^{6} b^{3} x^{11} + 3 \, a^{7} b^{2} x^{9} + 3 \, a^{8} b x^{7} + a^{9} x^{5}\right )}} - \frac {231 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{6}} \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
 

Output:

-1/240*(3465*b^5*x^10 + 9240*a*b^4*x^8 + 7623*a^2*b^3*x^6 + 1584*a^3*b^2*x 
^4 - 176*a^4*b*x^2 + 48*a^5)/(a^6*b^3*x^11 + 3*a^7*b^2*x^9 + 3*a^8*b*x^7 + 
 a^9*x^5) - 231/16*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {231 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{6}} - \frac {213 \, b^{5} x^{5} + 472 \, a b^{4} x^{3} + 267 \, a^{2} b^{3} x}{48 \, {\left (b x^{2} + a\right )}^{3} a^{6}} - \frac {150 \, b^{2} x^{4} - 20 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{6} x^{5}} \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
 

Output:

-231/16*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) - 1/48*(213*b^5*x^5 + 47 
2*a*b^4*x^3 + 267*a^2*b^3*x)/((b*x^2 + a)^3*a^6) - 1/15*(150*b^2*x^4 - 20* 
a*b*x^2 + 3*a^2)/(a^6*x^5)
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {1}{5\,a}-\frac {11\,b\,x^2}{15\,a^2}+\frac {33\,b^2\,x^4}{5\,a^3}+\frac {2541\,b^3\,x^6}{80\,a^4}+\frac {77\,b^4\,x^8}{2\,a^5}+\frac {231\,b^5\,x^{10}}{16\,a^6}}{a^3\,x^5+3\,a^2\,b\,x^7+3\,a\,b^2\,x^9+b^3\,x^{11}}-\frac {231\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,a^{13/2}} \] Input:

int(1/(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)^2),x)
 

Output:

- (1/(5*a) - (11*b*x^2)/(15*a^2) + (33*b^2*x^4)/(5*a^3) + (2541*b^3*x^6)/( 
80*a^4) + (77*b^4*x^8)/(2*a^5) + (231*b^5*x^10)/(16*a^6))/(a^3*x^5 + b^3*x 
^11 + 3*a^2*b*x^7 + 3*a*b^2*x^9) - (231*b^(5/2)*atan((b^(1/2)*x)/a^(1/2))) 
/(16*a^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.66 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{5}-10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{7}-10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{9}-3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{11}-48 a^{6}+176 a^{5} b \,x^{2}-1584 a^{4} b^{2} x^{4}-7623 a^{3} b^{3} x^{6}-9240 a^{2} b^{4} x^{8}-3465 a \,b^{5} x^{10}}{240 a^{7} x^{5} \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:

int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
 

Output:

( - 3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**5 - 10 
395*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*x**7 - 10395*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*x**9 - 3465*sqrt(b)*sq 
rt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*x**11 - 48*a**6 + 176*a**5*b*x**2 
 - 1584*a**4*b**2*x**4 - 7623*a**3*b**3*x**6 - 9240*a**2*b**4*x**8 - 3465* 
a*b**5*x**10)/(240*a**7*x**5*(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3* 
x**6))