Integrand size = 24, antiderivative size = 144 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1001}{256 a^6 x^3}+\frac {3003 b}{256 a^7 x}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {3003 b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{15/2}} \]
-1001/256/a^6/x^3+3003/256*b/a^7/x+1/10/a/x^3/(b*x^2+a)^5+13/80/a^2/x^3/(b *x^2+a)^4+143/480/a^3/x^3/(b*x^2+a)^3+429/640/a^4/x^3/(b*x^2+a)^2+3003/128 0/a^5/x^3/(b*x^2+a)+3003/256*b^(3/2)*arctan(x*b^(1/2)/a^(1/2))/a^(15/2)
Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {\sqrt {a} \left (-1280 a^6+16640 a^5 b x^2+137995 a^4 b^2 x^4+338910 a^3 b^3 x^6+384384 a^2 b^4 x^8+210210 a b^5 x^{10}+45045 b^6 x^{12}\right )}{x^3 \left (a+b x^2\right )^5}+45045 b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3840 a^{15/2}} \]
((Sqrt[a]*(-1280*a^6 + 16640*a^5*b*x^2 + 137995*a^4*b^2*x^4 + 338910*a^3*b ^3*x^6 + 384384*a^2*b^4*x^8 + 210210*a*b^5*x^10 + 45045*b^6*x^12))/(x^3*(a + b*x^2)^5) + 45045*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*a^(15/2))
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1380, 27, 253, 253, 253, 253, 253, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 x^4 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 \int \frac {1}{x^4 \left (b x^2+a\right )^5}dx}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 \left (\frac {11 \int \frac {1}{x^4 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \int \frac {1}{x^4 \left (b x^2+a\right )^3}dx}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {1}{x^4 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\) |
1/(10*a*x^3*(a + b*x^2)^5) + (13*(1/(8*a*x^3*(a + b*x^2)^4) + (11*(1/(6*a* x^3*(a + b*x^2)^3) + (3*(1/(4*a*x^3*(a + b*x^2)^2) + (7*(1/(2*a*x^3*(a + b *x^2)) + (5*(-1/3*1/(a*x^3) - (b*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x) /Sqrt[a]])/a^(3/2)))/a))/(2*a)))/(4*a)))/(2*a)))/(8*a)))/(10*a)
3.6.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 0.15 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(-\frac {1}{3 a^{6} x^{3}}+\frac {6 b}{a^{7} x}+\frac {b^{2} \left (\frac {\frac {1467}{256} b^{4} x^{9}+\frac {9629}{384} a \,b^{3} x^{7}+\frac {1253}{30} a^{2} b^{2} x^{5}+\frac {12131}{384} a^{3} b \,x^{3}+\frac {2373}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {3003 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{7}}\) | \(97\) |
| risch | \(\frac {\frac {3003 b^{6} x^{12}}{256 a^{7}}+\frac {7007 b^{5} x^{10}}{128 a^{6}}+\frac {1001 b^{4} x^{8}}{10 a^{5}}+\frac {11297 b^{3} x^{6}}{128 a^{4}}+\frac {27599 b^{2} x^{4}}{768 a^{3}}+\frac {13 b \,x^{2}}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {3003 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{15} \textit {\_Z}^{2}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{15}+2 b^{3}\right ) x -a^{8} b \textit {\_R} \right )\right )}{512}\) | \(149\) |
-1/3/a^6/x^3+6*b/a^7/x+b^2/a^7*((1467/256*b^4*x^9+9629/384*a*b^3*x^7+1253/ 30*a^2*b^2*x^5+12131/384*a^3*b*x^3+2373/256*a^4*x)/(b*x^2+a)^5+3003/256/(a *b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.03 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {90090 \, b^{6} x^{12} + 420420 \, a b^{5} x^{10} + 768768 \, a^{2} b^{4} x^{8} + 677820 \, a^{3} b^{3} x^{6} + 275990 \, a^{4} b^{2} x^{4} + 33280 \, a^{5} b x^{2} - 2560 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}, \frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}\right ] \]
[1/7680*(90090*b^6*x^12 + 420420*a*b^5*x^10 + 768768*a^2*b^4*x^8 + 677820* a^3*b^3*x^6 + 275990*a^4*b^2*x^4 + 33280*a^5*b*x^2 - 2560*a^6 + 45045*(b^6 *x^13 + 5*a*b^5*x^11 + 10*a^2*b^4*x^9 + 10*a^3*b^3*x^7 + 5*a^4*b^2*x^5 + a ^5*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^7 *b^5*x^13 + 5*a^8*b^4*x^11 + 10*a^9*b^3*x^9 + 10*a^10*b^2*x^7 + 5*a^11*b*x ^5 + a^12*x^3), 1/3840*(45045*b^6*x^12 + 210210*a*b^5*x^10 + 384384*a^2*b^ 4*x^8 + 338910*a^3*b^3*x^6 + 137995*a^4*b^2*x^4 + 16640*a^5*b*x^2 - 1280*a ^6 + 45045*(b^6*x^13 + 5*a*b^5*x^11 + 10*a^2*b^4*x^9 + 10*a^3*b^3*x^7 + 5* a^4*b^2*x^5 + a^5*b*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^7*b^5*x^13 + 5* a^8*b^4*x^11 + 10*a^9*b^3*x^9 + 10*a^10*b^2*x^7 + 5*a^11*b*x^5 + a^12*x^3) ]
Time = 0.42 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (- \frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (\frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {- 1280 a^{6} + 16640 a^{5} b x^{2} + 137995 a^{4} b^{2} x^{4} + 338910 a^{3} b^{3} x^{6} + 384384 a^{2} b^{4} x^{8} + 210210 a b^{5} x^{10} + 45045 b^{6} x^{12}}{3840 a^{12} x^{3} + 19200 a^{11} b x^{5} + 38400 a^{10} b^{2} x^{7} + 38400 a^{9} b^{3} x^{9} + 19200 a^{8} b^{4} x^{11} + 3840 a^{7} b^{5} x^{13}} \]
-3003*sqrt(-b**3/a**15)*log(-a**8*sqrt(-b**3/a**15)/b**2 + x)/512 + 3003*s qrt(-b**3/a**15)*log(a**8*sqrt(-b**3/a**15)/b**2 + x)/512 + (-1280*a**6 + 16640*a**5*b*x**2 + 137995*a**4*b**2*x**4 + 338910*a**3*b**3*x**6 + 384384 *a**2*b**4*x**8 + 210210*a*b**5*x**10 + 45045*b**6*x**12)/(3840*a**12*x**3 + 19200*a**11*b*x**5 + 38400*a**10*b**2*x**7 + 38400*a**9*b**3*x**9 + 192 00*a**8*b**4*x**11 + 3840*a**7*b**5*x**13)
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6}}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}} + \frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} \]
1/3840*(45045*b^6*x^12 + 210210*a*b^5*x^10 + 384384*a^2*b^4*x^8 + 338910*a ^3*b^3*x^6 + 137995*a^4*b^2*x^4 + 16640*a^5*b*x^2 - 1280*a^6)/(a^7*b^5*x^1 3 + 5*a^8*b^4*x^11 + 10*a^9*b^3*x^9 + 10*a^10*b^2*x^7 + 5*a^11*b*x^5 + a^1 2*x^3) + 3003/256*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7)
Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} + \frac {18 \, b x^{2} - a}{3 \, a^{7} x^{3}} + \frac {22005 \, b^{6} x^{9} + 96290 \, a b^{5} x^{7} + 160384 \, a^{2} b^{4} x^{5} + 121310 \, a^{3} b^{3} x^{3} + 35595 \, a^{4} b^{2} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a^{7}} \]
3003/256*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7) + 1/3*(18*b*x^2 - a)/(a ^7*x^3) + 1/3840*(22005*b^6*x^9 + 96290*a*b^5*x^7 + 160384*a^2*b^4*x^5 + 1 21310*a^3*b^3*x^3 + 35595*a^4*b^2*x)/((b*x^2 + a)^5*a^7)
Time = 13.61 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {13\,b\,x^2}{3\,a^2}-\frac {1}{3\,a}+\frac {27599\,b^2\,x^4}{768\,a^3}+\frac {11297\,b^3\,x^6}{128\,a^4}+\frac {1001\,b^4\,x^8}{10\,a^5}+\frac {7007\,b^5\,x^{10}}{128\,a^6}+\frac {3003\,b^6\,x^{12}}{256\,a^7}}{a^5\,x^3+5\,a^4\,b\,x^5+10\,a^3\,b^2\,x^7+10\,a^2\,b^3\,x^9+5\,a\,b^4\,x^{11}+b^5\,x^{13}}+\frac {3003\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{15/2}} \]
((13*b*x^2)/(3*a^2) - 1/(3*a) + (27599*b^2*x^4)/(768*a^3) + (11297*b^3*x^6 )/(128*a^4) + (1001*b^4*x^8)/(10*a^5) + (7007*b^5*x^10)/(128*a^6) + (3003* b^6*x^12)/(256*a^7))/(a^5*x^3 + b^5*x^13 + 5*a^4*b*x^5 + 5*a*b^4*x^11 + 10 *a^3*b^2*x^7 + 10*a^2*b^3*x^9) + (3003*b^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/ (256*a^(15/2))