\(\int \frac {1}{x^6 (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [418]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1}{5 a^6 x^5}+\frac {2 b}{a^7 x^3}-\frac {21 b^2}{a^8 x}-\frac {b^3 x}{10 a^4 \left (a+b x^2\right )^5}-\frac {39 b^3 x}{80 a^5 \left (a+b x^2\right )^4}-\frac {251 b^3 x}{160 a^6 \left (a+b x^2\right )^3}-\frac {571 b^3 x}{128 a^7 \left (a+b x^2\right )^2}-\frac {3633 b^3 x}{256 a^8 \left (a+b x^2\right )}-\frac {9009 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}} \] Output:

-1/5/a^6/x^5+2*b/a^7/x^3-21*b^2/a^8/x-1/10*b^3*x/a^4/(b*x^2+a)^5-39/80*b^3 
*x/a^5/(b*x^2+a)^4-251/160*b^3*x/a^6/(b*x^2+a)^3-571/128*b^3*x/a^7/(b*x^2+ 
a)^2-3633/256*b^3*x/a^8/(b*x^2+a)-9009/256*b^(5/2)*arctan(b^(1/2)*x/a^(1/2 
))/a^(17/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {256 a^7-1280 a^6 b x^2+16640 a^5 b^2 x^4+137995 a^4 b^3 x^6+338910 a^3 b^4 x^8+384384 a^2 b^5 x^{10}+210210 a b^6 x^{12}+45045 b^7 x^{14}}{1280 a^8 x^5 \left (a+b x^2\right )^5}-\frac {9009 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{17/2}} \] Input:

Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
 

Output:

-1/1280*(256*a^7 - 1280*a^6*b*x^2 + 16640*a^5*b^2*x^4 + 137995*a^4*b^3*x^6 
 + 338910*a^3*b^4*x^8 + 384384*a^2*b^5*x^10 + 210210*a*b^6*x^12 + 45045*b^ 
7*x^14)/(a^8*x^5*(a + b*x^2)^5) - (9009*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a] 
])/(256*a^(17/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1380, 27, 253, 253, 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.

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {3 \int \frac {1}{x^6 \left (b x^2+a\right )^5}dx}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {3 \left (\frac {13 \int \frac {1}{x^6 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \int \frac {1}{x^6 \left (b x^2+a\right )^3}dx}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \int \frac {1}{x^6 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{x^6 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\)

Input:

Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
 

Output:

1/(10*a*x^5*(a + b*x^2)^5) + (3*(1/(8*a*x^5*(a + b*x^2)^4) + (13*(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)))/ 
(8*a)))/(2*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 = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {b^{3} \left (\frac {\frac {3633}{256} b^{4} x^{9}+\frac {7837}{128} a \,b^{3} x^{7}+\frac {1001}{10} a^{2} b^{2} x^{5}+\frac {9443}{128} a^{3} b \,x^{3}+\frac {5327}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {9009 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{8}}-\frac {1}{5 a^{6} x^{5}}-\frac {21 b^{2}}{a^{8} x}+\frac {2 b}{a^{7} x^{3}}\)	\(109\)
risch	\(\frac {-\frac {1}{5 a}+\frac {b \,x^{2}}{a^{2}}-\frac {13 b^{2} x^{4}}{a^{3}}-\frac {27599 b^{3} x^{6}}{256 a^{4}}-\frac {33891 b^{4} x^{8}}{128 a^{5}}-\frac {3003 b^{5} x^{10}}{10 a^{6}}-\frac {21021 b^{6} x^{12}}{128 a^{7}}-\frac {9009 b^{7} x^{14}}{256 a^{8}}}{x^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {9009 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right )}{512 a^{9}}-\frac {9009 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right )}{512 a^{9}}\)	\(169\)

Input:

int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/a^8*b^3*((3633/256*b^4*x^9+7837/128*a*b^3*x^7+1001/10*a^2*b^2*x^5+9443/ 
128*a^3*b*x^3+5327/256*a^4*x)/(b*x^2+a)^5+9009/256/(a*b)^(1/2)*arctan(b/(a 
*b)^(1/2)*x))-1/5/a^6/x^5-21*b^2/a^8/x+2*b/a^7/x^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.92 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [-\frac {90090 \, b^{7} x^{14} + 420420 \, a b^{6} x^{12} + 768768 \, a^{2} b^{5} x^{10} + 677820 \, a^{3} b^{4} x^{8} + 275990 \, a^{4} b^{3} x^{6} + 33280 \, a^{5} b^{2} x^{4} - 2560 \, a^{6} b x^{2} + 512 \, a^{7} - 45045 \, {\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2560 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}}, -\frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7} + 45045 \, {\left (b^{7} x^{15} + 5 \, a b^{6} x^{13} + 10 \, a^{2} b^{5} x^{11} + 10 \, a^{3} b^{4} x^{9} + 5 \, a^{4} b^{3} x^{7} + a^{5} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{1280 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}}\right ] \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

[-1/2560*(90090*b^7*x^14 + 420420*a*b^6*x^12 + 768768*a^2*b^5*x^10 + 67782 
0*a^3*b^4*x^8 + 275990*a^4*b^3*x^6 + 33280*a^5*b^2*x^4 - 2560*a^6*b*x^2 + 
512*a^7 - 45045*(b^7*x^15 + 5*a*b^6*x^13 + 10*a^2*b^5*x^11 + 10*a^3*b^4*x^ 
9 + 5*a^4*b^3*x^7 + a^5*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) 
- a)/(b*x^2 + a)))/(a^8*b^5*x^15 + 5*a^9*b^4*x^13 + 10*a^10*b^3*x^11 + 10* 
a^11*b^2*x^9 + 5*a^12*b*x^7 + a^13*x^5), -1/1280*(45045*b^7*x^14 + 210210* 
a*b^6*x^12 + 384384*a^2*b^5*x^10 + 338910*a^3*b^4*x^8 + 137995*a^4*b^3*x^6 
 + 16640*a^5*b^2*x^4 - 1280*a^6*b*x^2 + 256*a^7 + 45045*(b^7*x^15 + 5*a*b^ 
6*x^13 + 10*a^2*b^5*x^11 + 10*a^3*b^4*x^9 + 5*a^4*b^3*x^7 + a^5*b^2*x^5)*s 
qrt(b/a)*arctan(x*sqrt(b/a)))/(a^8*b^5*x^15 + 5*a^9*b^4*x^13 + 10*a^10*b^3 
*x^11 + 10*a^11*b^2*x^9 + 5*a^12*b*x^7 + a^13*x^5)]
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {9009 \sqrt {- \frac {b^{5}}{a^{17}}} \log {\left (- \frac {a^{9} \sqrt {- \frac {b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} - \frac {9009 \sqrt {- \frac {b^{5}}{a^{17}}} \log {\left (\frac {a^{9} \sqrt {- \frac {b^{5}}{a^{17}}}}{b^{3}} + x \right )}}{512} + \frac {- 256 a^{7} + 1280 a^{6} b x^{2} - 16640 a^{5} b^{2} x^{4} - 137995 a^{4} b^{3} x^{6} - 338910 a^{3} b^{4} x^{8} - 384384 a^{2} b^{5} x^{10} - 210210 a b^{6} x^{12} - 45045 b^{7} x^{14}}{1280 a^{13} x^{5} + 6400 a^{12} b x^{7} + 12800 a^{11} b^{2} x^{9} + 12800 a^{10} b^{3} x^{11} + 6400 a^{9} b^{4} x^{13} + 1280 a^{8} b^{5} x^{15}} \] Input:

integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

9009*sqrt(-b**5/a**17)*log(-a**9*sqrt(-b**5/a**17)/b**3 + x)/512 - 9009*sq 
rt(-b**5/a**17)*log(a**9*sqrt(-b**5/a**17)/b**3 + x)/512 + (-256*a**7 + 12 
80*a**6*b*x**2 - 16640*a**5*b**2*x**4 - 137995*a**4*b**3*x**6 - 338910*a** 
3*b**4*x**8 - 384384*a**2*b**5*x**10 - 210210*a*b**6*x**12 - 45045*b**7*x* 
*14)/(1280*a**13*x**5 + 6400*a**12*b*x**7 + 12800*a**11*b**2*x**9 + 12800* 
a**10*b**3*x**11 + 6400*a**9*b**4*x**13 + 1280*a**8*b**5*x**15)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7}}{1280 \, {\left (a^{8} b^{5} x^{15} + 5 \, a^{9} b^{4} x^{13} + 10 \, a^{10} b^{3} x^{11} + 10 \, a^{11} b^{2} x^{9} + 5 \, a^{12} b x^{7} + a^{13} x^{5}\right )}} - \frac {9009 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{8}} \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

-1/1280*(45045*b^7*x^14 + 210210*a*b^6*x^12 + 384384*a^2*b^5*x^10 + 338910 
*a^3*b^4*x^8 + 137995*a^4*b^3*x^6 + 16640*a^5*b^2*x^4 - 1280*a^6*b*x^2 + 2 
56*a^7)/(a^8*b^5*x^15 + 5*a^9*b^4*x^13 + 10*a^10*b^3*x^11 + 10*a^11*b^2*x^ 
9 + 5*a^12*b*x^7 + a^13*x^5) - 9009/256*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a* 
b)*a^8)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {9009 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{8}} - \frac {45045 \, b^{7} x^{14} + 210210 \, a b^{6} x^{12} + 384384 \, a^{2} b^{5} x^{10} + 338910 \, a^{3} b^{4} x^{8} + 137995 \, a^{4} b^{3} x^{6} + 16640 \, a^{5} b^{2} x^{4} - 1280 \, a^{6} b x^{2} + 256 \, a^{7}}{1280 \, {\left (b x^{3} + a x\right )}^{5} a^{8}} \] Input:

integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

-9009/256*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8) - 1/1280*(45045*b^7*x^ 
14 + 210210*a*b^6*x^12 + 384384*a^2*b^5*x^10 + 338910*a^3*b^4*x^8 + 137995 
*a^4*b^3*x^6 + 16640*a^5*b^2*x^4 - 1280*a^6*b*x^2 + 256*a^7)/((b*x^3 + a*x 
)^5*a^8)
 

Mupad [B] (verification not implemented)

Time = 18.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {1}{5\,a}-\frac {b\,x^2}{a^2}+\frac {13\,b^2\,x^4}{a^3}+\frac {27599\,b^3\,x^6}{256\,a^4}+\frac {33891\,b^4\,x^8}{128\,a^5}+\frac {3003\,b^5\,x^{10}}{10\,a^6}+\frac {21021\,b^6\,x^{12}}{128\,a^7}+\frac {9009\,b^7\,x^{14}}{256\,a^8}}{a^5\,x^5+5\,a^4\,b\,x^7+10\,a^3\,b^2\,x^9+10\,a^2\,b^3\,x^{11}+5\,a\,b^4\,x^{13}+b^5\,x^{15}}-\frac {9009\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{17/2}} \] Input:

int(1/(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)
 

Output:

- (1/(5*a) - (b*x^2)/a^2 + (13*b^2*x^4)/a^3 + (27599*b^3*x^6)/(256*a^4) + 
(33891*b^4*x^8)/(128*a^5) + (3003*b^5*x^10)/(10*a^6) + (21021*b^6*x^12)/(1 
28*a^7) + (9009*b^7*x^14)/(256*a^8))/(a^5*x^5 + b^5*x^15 + 5*a^4*b*x^7 + 5 
*a*b^4*x^13 + 10*a^3*b^2*x^9 + 10*a^2*b^3*x^11) - (9009*b^(5/2)*atan((b^(1 
/2)*x)/a^(1/2)))/(256*a^(17/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {-45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{2} x^{5}-225225 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{3} x^{7}-450450 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{4} x^{9}-450450 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{5} x^{11}-225225 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{6} x^{13}-45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{7} x^{15}-256 a^{8}+1280 a^{7} b \,x^{2}-16640 a^{6} b^{2} x^{4}-137995 a^{5} b^{3} x^{6}-338910 a^{4} b^{4} x^{8}-384384 a^{3} b^{5} x^{10}-210210 a^{2} b^{6} x^{12}-45045 a \,b^{7} x^{14}}{1280 a^{9} x^{5} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:

int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

( - 45045*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**2*x**5 - 2 
25225*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**3*x**7 - 45045 
0*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**4*x**9 - 450450*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**5*x**11 - 225225*sqrt( 
b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**6*x**13 - 45045*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**7*x**15 - 256*a**8 + 1280*a**7*b*x**2 
 - 16640*a**6*b**2*x**4 - 137995*a**5*b**3*x**6 - 338910*a**4*b**4*x**8 - 
384384*a**3*b**5*x**10 - 210210*a**2*b**6*x**12 - 45045*a*b**7*x**14)/(128 
0*a**9*x**5*(a**5 + 5*a**4*b*x**2 + 10*a**3*b**2*x**4 + 10*a**2*b**3*x**6 
+ 5*a*b**4*x**8 + b**5*x**10))