\(\int \frac {x^{10}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [410]

Optimal result

Integrand size = 24, antiderivative size = 121 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac {63 x}{256 b^5 \left (a+b x^2\right )}+\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}} \] Output:

-1/10*x^9/b/(b*x^2+a)^5-9/80*x^7/b^2/(b*x^2+a)^4-21/160*x^5/b^3/(b*x^2+a)^ 
3-21/128*x^3/b^4/(b*x^2+a)^2-63/256*x/b^5/(b*x^2+a)+63/256*arctan(b^(1/2)* 
x/a^(1/2))/a^(1/2)/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x \left (315 a^4+1470 a^3 b x^2+2688 a^2 b^2 x^4+2370 a b^3 x^6+965 b^4 x^8\right )}{1280 b^5 \left (a+b x^2\right )^5}+\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}} \] Input:

Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/1280*(x*(315*a^4 + 1470*a^3*b*x^2 + 2688*a^2*b^2*x^4 + 2370*a*b^3*x^6 + 
 965*b^4*x^8))/(b^5*(a + b*x^2)^5) + (63*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256 
*Sqrt[a]*b^(11/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.26, 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, 252, 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.

\(\displaystyle \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 \int \frac {x^8}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x^9}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 218

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

Input:

Int[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/10*x^9/(b*(a + b*x^2)^5) + (9*(-1/8*x^7/(b*(a + b*x^2)^4) + (7*(-1/6*x^ 
5/(b*(a + b*x^2)^3) + (5*(-1/4*x^3/(b*(a + b*x^2)^2) + (3*(-1/2*x/(b*(a + 
b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))))/(4*b)))/(6*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 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 = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {-\frac {193 x^{9}}{256 b}-\frac {237 a \,x^{7}}{128 b^{2}}-\frac {21 a^{2} x^{5}}{10 b^{3}}-\frac {147 a^{3} x^{3}}{128 b^{4}}-\frac {63 a^{4} x}{256 b^{5}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {63 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 b^{5} \sqrt {a b}}\)	\(80\)
risch	\(\frac {-\frac {193 x^{9}}{256 b}-\frac {237 a \,x^{7}}{128 b^{2}}-\frac {21 a^{2} x^{5}}{10 b^{3}}-\frac {147 a^{3} x^{3}}{128 b^{4}}-\frac {63 a^{4} x}{256 b^{5}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}-\frac {63 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{5}}+\frac {63 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{5}}\)	\(126\)

Input:

int(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

(-193/256/b*x^9-237/128*a/b^2*x^7-21/10*a^2/b^3*x^5-147/128*a^3/b^4*x^3-63 
/256*a^4/b^5*x)/(b*x^2+a)^5+63/256/b^5/(a*b)^(1/2)*arctan(b/(a*b)^(1/2)*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.19 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [-\frac {1930 \, a b^{5} x^{9} + 4740 \, a^{2} b^{4} x^{7} + 5376 \, a^{3} b^{3} x^{5} + 2940 \, a^{4} b^{2} x^{3} + 630 \, a^{5} b x + 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2560 \, {\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}, -\frac {965 \, a b^{5} x^{9} + 2370 \, a^{2} b^{4} x^{7} + 2688 \, a^{3} b^{3} x^{5} + 1470 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x - 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{1280 \, {\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}\right ] \] Input:

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

[-1/2560*(1930*a*b^5*x^9 + 4740*a^2*b^4*x^7 + 5376*a^3*b^3*x^5 + 2940*a^4* 
b^2*x^3 + 630*a^5*b*x + 315*(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*b^11*x^10 + 5*a^2*b^10*x^8 + 10*a^3*b^9*x^6 + 10*a^4*b 
^8*x^4 + 5*a^5*b^7*x^2 + a^6*b^6), -1/1280*(965*a*b^5*x^9 + 2370*a^2*b^4*x 
^7 + 2688*a^3*b^3*x^5 + 1470*a^4*b^2*x^3 + 315*a^5*b*x - 315*(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 
)*arctan(sqrt(a*b)*x/a))/(a*b^11*x^10 + 5*a^2*b^10*x^8 + 10*a^3*b^9*x^6 + 
10*a^4*b^8*x^4 + 5*a^5*b^7*x^2 + a^6*b^6)]
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.50 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {63 \sqrt {- \frac {1}{a b^{11}}} \log {\left (- a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{512} + \frac {63 \sqrt {- \frac {1}{a b^{11}}} \log {\left (a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{512} + \frac {- 315 a^{4} x - 1470 a^{3} b x^{3} - 2688 a^{2} b^{2} x^{5} - 2370 a b^{3} x^{7} - 965 b^{4} x^{9}}{1280 a^{5} b^{5} + 6400 a^{4} b^{6} x^{2} + 12800 a^{3} b^{7} x^{4} + 12800 a^{2} b^{8} x^{6} + 6400 a b^{9} x^{8} + 1280 b^{10} x^{10}} \] Input:

integrate(x**10/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

-63*sqrt(-1/(a*b**11))*log(-a*b**5*sqrt(-1/(a*b**11)) + x)/512 + 63*sqrt(- 
1/(a*b**11))*log(a*b**5*sqrt(-1/(a*b**11)) + x)/512 + (-315*a**4*x - 1470* 
a**3*b*x**3 - 2688*a**2*b**2*x**5 - 2370*a*b**3*x**7 - 965*b**4*x**9)/(128 
0*a**5*b**5 + 6400*a**4*b**6*x**2 + 12800*a**3*b**7*x**4 + 12800*a**2*b**8 
*x**6 + 6400*a*b**9*x**8 + 1280*b**10*x**10)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {965 \, b^{4} x^{9} + 2370 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 1470 \, a^{3} b x^{3} + 315 \, a^{4} x}{1280 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} + \frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{5}} \] Input:

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

-1/1280*(965*b^4*x^9 + 2370*a*b^3*x^7 + 2688*a^2*b^2*x^5 + 1470*a^3*b*x^3 
+ 315*a^4*x)/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 
5*a^4*b^6*x^2 + a^5*b^5) + 63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{5}} - \frac {965 \, b^{4} x^{9} + 2370 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 1470 \, a^{3} b x^{3} + 315 \, a^{4} x}{1280 \, {\left (b x^{2} + a\right )}^{5} b^{5}} \] Input:

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/1280*(965*b^4*x^9 + 2370* 
a*b^3*x^7 + 2688*a^2*b^2*x^5 + 1470*a^3*b*x^3 + 315*a^4*x)/((b*x^2 + a)^5* 
b^5)
 

Mupad [B] (verification not implemented)

Time = 18.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,\sqrt {a}\,b^{11/2}}-\frac {\frac {193\,x^9}{256\,b}+\frac {237\,a\,x^7}{128\,b^2}+\frac {63\,a^4\,x}{256\,b^5}+\frac {21\,a^2\,x^5}{10\,b^3}+\frac {147\,a^3\,x^3}{128\,b^4}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \] Input:

int(x^10/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

(63*atan((b^(1/2)*x)/a^(1/2)))/(256*a^(1/2)*b^(11/2)) - ((193*x^9)/(256*b) 
 + (237*a*x^7)/(128*b^2) + (63*a^4*x)/(256*b^5) + (21*a^2*x^5)/(10*b^3) + 
(147*a^3*x^3)/(128*b^4))/(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 = 2.15 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5}+1575 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b \,x^{2}+3150 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{4}+3150 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{6}+1575 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{8}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{10}-315 a^{5} b x -1470 a^{4} b^{2} x^{3}-2688 a^{3} b^{3} x^{5}-2370 a^{2} b^{4} x^{7}-965 a \,b^{5} x^{9}}{1280 a \,b^{6} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:

int(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5 + 1575*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*x**2 + 3150*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*x**4 + 3150*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**2*b**3*x**6 + 1575*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a*b**4*x**8 + 315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*b**5*x**10 - 315*a**5*b*x - 1470*a**4*b**2*x**3 - 2688*a**3*b**3*x**5 
- 2370*a**2*b**4*x**7 - 965*a*b**5*x**9)/(1280*a*b**6*(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))