\(\int \frac {x^8}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [411]

Optimal result

Integrand size = 24, antiderivative size = 122 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac {7 x}{256 a b^4 \left (a+b x^2\right )}+\frac {7 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}} \] Output:

-1/10*x^7/b/(b*x^2+a)^5-7/80*x^5/b^2/(b*x^2+a)^4-7/96*x^3/b^3/(b*x^2+a)^3- 
7/128*x/b^4/(b*x^2+a)^2+7/256*x/a/b^4/(b*x^2+a)+7/256*arctan(b^(1/2)*x/a^( 
1/2))/a^(3/2)/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x \left (105 a^4+490 a^3 b x^2+896 a^2 b^2 x^4+790 a b^3 x^6-105 b^4 x^8\right )}{3840 a b^4 \left (a+b x^2\right )^5}+\frac {7 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}} \] Input:

Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/3840*(x*(105*a^4 + 490*a^3*b*x^2 + 896*a^2*b^2*x^4 + 790*a*b^3*x^6 - 10 
5*b^4*x^8))/(a*b^4*(a + b*x^2)^5) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a 
^(3/2)*b^(9/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1380, 27, 252, 252, 252, 252, 215, 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.

\(\displaystyle \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^6 \int \frac {x^8}{b^6 \left (b x^2+a\right )^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^6}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 \int \frac {x^6}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x^7}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^5}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^7}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

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

Input:

Int[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/10*x^7/(b*(a + b*x^2)^5) + (7*(-1/8*x^5/(b*(a + b*x^2)^4) + (5*(-1/6*x^ 
3/(b*(a + b*x^2)^3) + (-1/4*x/(b*(a + b*x^2)^2) + (x/(2*a*(a + b*x^2)) + A 
rcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]))/(4*b))/(2*b)))/(8*b)))/(10 
*b)
 

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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[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 = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\frac {7 x^{9}}{256 a}-\frac {79 x^{7}}{384 b}-\frac {7 a \,x^{5}}{30 b^{2}}-\frac {49 a^{2} x^{3}}{384 b^{3}}-\frac {7 a^{3} x}{256 b^{4}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 b^{4} a \sqrt {a b}}\)	\(80\)
risch	\(\frac {\frac {7 x^{9}}{256 a}-\frac {79 x^{7}}{384 b}-\frac {7 a \,x^{5}}{30 b^{2}}-\frac {49 a^{2} x^{3}}{384 b^{3}}-\frac {7 a^{3} x}{256 b^{4}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}-\frac {7 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{4} a}+\frac {7 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{4} a}\)	\(129\)

Input:

int(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

(7/256/a*x^9-79/384/b*x^7-7/30*a/b^2*x^5-49/384*a^2/b^3*x^3-7/256*a^3/b^4* 
x)/(b*x^2+a)^5+7/256/b^4/a/(a*b)^(1/2)*arctan(b/(a*b)^(1/2)*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.20 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {210 \, a b^{5} x^{9} - 1580 \, a^{2} b^{4} x^{7} - 1792 \, a^{3} b^{3} x^{5} - 980 \, a^{4} b^{2} x^{3} - 210 \, a^{5} b x - 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, \frac {105 \, a b^{5} x^{9} - 790 \, a^{2} b^{4} x^{7} - 896 \, a^{3} b^{3} x^{5} - 490 \, a^{4} b^{2} x^{3} - 105 \, a^{5} b x + 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{3840 \, {\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

[1/7680*(210*a*b^5*x^9 - 1580*a^2*b^4*x^7 - 1792*a^3*b^3*x^5 - 980*a^4*b^2 
*x^3 - 210*a^5*b*x - 105*(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)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/ 
(b*x^2 + a)))/(a^2*b^10*x^10 + 5*a^3*b^9*x^8 + 10*a^4*b^8*x^6 + 10*a^5*b^7 
*x^4 + 5*a^6*b^6*x^2 + a^7*b^5), 1/3840*(105*a*b^5*x^9 - 790*a^2*b^4*x^7 - 
 896*a^3*b^3*x^5 - 490*a^4*b^2*x^3 - 105*a^5*b*x + 105*(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)*sqrt(a*b)*arct 
an(sqrt(a*b)*x/a))/(a^2*b^10*x^10 + 5*a^3*b^9*x^8 + 10*a^4*b^8*x^6 + 10*a^ 
5*b^7*x^4 + 5*a^6*b^6*x^2 + a^7*b^5)]
 

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.59 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {7 \sqrt {- \frac {1}{a^{3} b^{9}}} \log {\left (- a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} + x \right )}}{512} + \frac {7 \sqrt {- \frac {1}{a^{3} b^{9}}} \log {\left (a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} + x \right )}}{512} + \frac {- 105 a^{4} x - 490 a^{3} b x^{3} - 896 a^{2} b^{2} x^{5} - 790 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{6} b^{4} + 19200 a^{5} b^{5} x^{2} + 38400 a^{4} b^{6} x^{4} + 38400 a^{3} b^{7} x^{6} + 19200 a^{2} b^{8} x^{8} + 3840 a b^{9} x^{10}} \] Input:

integrate(x**8/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

-7*sqrt(-1/(a**3*b**9))*log(-a**2*b**4*sqrt(-1/(a**3*b**9)) + x)/512 + 7*s 
qrt(-1/(a**3*b**9))*log(a**2*b**4*sqrt(-1/(a**3*b**9)) + x)/512 + (-105*a* 
*4*x - 490*a**3*b*x**3 - 896*a**2*b**2*x**5 - 790*a*b**3*x**7 + 105*b**4*x 
**9)/(3840*a**6*b**4 + 19200*a**5*b**5*x**2 + 38400*a**4*b**6*x**4 + 38400 
*a**3*b**7*x**6 + 19200*a**2*b**8*x**8 + 3840*a*b**9*x**10)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} + \frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a b^{4}} \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

1/3840*(105*b^4*x^9 - 790*a*b^3*x^7 - 896*a^2*b^2*x^5 - 490*a^3*b*x^3 - 10 
5*a^4*x)/(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5 
*a^5*b^5*x^2 + a^6*b^4) + 7/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^4)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.69 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a b^{4}} + \frac {105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a b^{4}} \] Input:

integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

7/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^4) + 1/3840*(105*b^4*x^9 - 790* 
a*b^3*x^7 - 896*a^2*b^2*x^5 - 490*a^3*b*x^3 - 105*a^4*x)/((b*x^2 + a)^5*a* 
b^4)
 

Mupad [B] (verification not implemented)

Time = 18.81 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{3/2}\,b^{9/2}}-\frac {\frac {79\,x^7}{384\,b}-\frac {7\,x^9}{256\,a}+\frac {7\,a\,x^5}{30\,b^2}+\frac {7\,a^3\,x}{256\,b^4}+\frac {49\,a^2\,x^3}{384\,b^3}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \] Input:

int(x^8/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

(7*atan((b^(1/2)*x)/a^(1/2)))/(256*a^(3/2)*b^(9/2)) - ((79*x^7)/(384*b) - 
(7*x^9)/(256*a) + (7*a*x^5)/(30*b^2) + (7*a^3*x)/(256*b^4) + (49*a^2*x^3)/ 
(384*b^3))/(a^5 + b^5*x^10 + 5*a^4*b*x^2 + 5*a*b^4*x^8 + 10*a^3*b^2*x^4 + 
10*a^2*b^3*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.13 \[ \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5}+525 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{2}+1050 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{4}+1050 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{6}+525 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{8}+105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{10}-105 a^{5} b x -490 a^{4} b^{2} x^{3}-896 a^{3} b^{3} x^{5}-790 a^{2} b^{4} x^{7}+105 a \,b^{5} x^{9}}{3840 a^{2} b^{5} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:

int(x^8/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5 + 525*sqrt(b)*sqrt 
(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**2 + 1050*sqrt(b)*sqrt(a)*atan( 
(b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**4 + 1050*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**3*x**6 + 525*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a*b**4*x**8 + 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)) 
)*b**5*x**10 - 105*a**5*b*x - 490*a**4*b**2*x**3 - 896*a**3*b**3*x**5 - 79 
0*a**2*b**4*x**7 + 105*a*b**5*x**9)/(3840*a**2*b**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))