Integrand size = 24, antiderivative size = 140 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1}{4 a^6 x^4}+\frac {3 b}{a^7 x^2}+\frac {b^2}{10 a^3 \left (a+b x^2\right )^5}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{a^5 \left (a+b x^2\right )^3}+\frac {5 b^2}{2 a^6 \left (a+b x^2\right )^2}+\frac {15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac {21 b^2 \log (x)}{a^8}-\frac {21 b^2 \log \left (a+b x^2\right )}{2 a^8} \] Output:
-1/4/a^6/x^4+3*b/a^7/x^2+1/10*b^2/a^3/(b*x^2+a)^5+3/8*b^2/a^4/(b*x^2+a)^4+ b^2/a^5/(b*x^2+a)^3+5/2*b^2/a^6/(b*x^2+a)^2+15/2*b^2/a^7/(b*x^2+a)+21*b^2* ln(x)/a^8-21/2*b^2*ln(b*x^2+a)/a^8
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {a \left (-10 a^6+70 a^5 b x^2+959 a^4 b^2 x^4+2695 a^3 b^3 x^6+3290 a^2 b^4 x^8+1890 a b^5 x^{10}+420 b^6 x^{12}\right )}{x^4 \left (a+b x^2\right )^5}+840 b^2 \log (x)-420 b^2 \log \left (a+b x^2\right )}{40 a^8} \] Input:
Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
((a*(-10*a^6 + 70*a^5*b*x^2 + 959*a^4*b^2*x^4 + 2695*a^3*b^3*x^6 + 3290*a^ 2*b^4*x^8 + 1890*a*b^5*x^10 + 420*b^6*x^12))/(x^4*(a + b*x^2)^5) + 840*b^2 *Log[x] - 420*b^2*Log[a + b*x^2])/(40*a^8)
Time = 0.52 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1380, 27, 243, 54, 2009}
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.
| \(\displaystyle \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^6 \int \frac {1}{b^6 x^5 \left (b x^2+a\right )^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{x^5 \left (a+b x^2\right )^6}dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (b x^2+a\right )^6}dx^2\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {21 b^3}{a^8 \left (b x^2+a\right )}-\frac {15 b^3}{a^7 \left (b x^2+a\right )^2}-\frac {10 b^3}{a^6 \left (b x^2+a\right )^3}-\frac {6 b^3}{a^5 \left (b x^2+a\right )^4}-\frac {3 b^3}{a^4 \left (b x^2+a\right )^5}-\frac {b^3}{a^3 \left (b x^2+a\right )^6}+\frac {21 b^2}{a^8 x^2}-\frac {6 b}{a^7 x^4}+\frac {1}{a^6 x^6}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {21 b^2 \log \left (x^2\right )}{a^8}-\frac {21 b^2 \log \left (a+b x^2\right )}{a^8}+\frac {15 b^2}{a^7 \left (a+b x^2\right )}+\frac {6 b}{a^7 x^2}+\frac {5 b^2}{a^6 \left (a+b x^2\right )^2}-\frac {1}{2 a^6 x^4}+\frac {2 b^2}{a^5 \left (a+b x^2\right )^3}+\frac {3 b^2}{4 a^4 \left (a+b x^2\right )^4}+\frac {b^2}{5 a^3 \left (a+b x^2\right )^5}\right )\) |
Input:
Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]
Output:
(-1/2*1/(a^6*x^4) + (6*b)/(a^7*x^2) + b^2/(5*a^3*(a + b*x^2)^5) + (3*b^2)/ (4*a^4*(a + b*x^2)^4) + (2*b^2)/(a^5*(a + b*x^2)^3) + (5*b^2)/(a^6*(a + b* x^2)^2) + (15*b^2)/(a^7*(a + b*x^2)) + (21*b^2*Log[x^2])/a^8 - (21*b^2*Log [a + b*x^2])/a^8)/2
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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 = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79
| method | result | size |
| norman | \(\frac {-\frac {1}{4 a}+\frac {7 b \,x^{2}}{4 a^{2}}-\frac {105 b^{3} x^{6}}{2 a^{4}}-\frac {315 b^{4} x^{8}}{2 a^{5}}-\frac {385 b^{5} x^{10}}{2 a^{6}}-\frac {875 b^{6} x^{12}}{8 a^{7}}-\frac {959 b^{7} x^{14}}{40 a^{8}}}{x^{4} \left (b \,x^{2}+a \right )^{5}}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}-\frac {21 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{8}}\) | \(111\) |
| risch | \(\frac {\frac {21 b^{6} x^{12}}{2 a^{7}}+\frac {189 b^{5} x^{10}}{4 a^{6}}+\frac {329 b^{4} x^{8}}{4 a^{5}}+\frac {539 b^{3} x^{6}}{8 a^{4}}+\frac {959 b^{2} x^{4}}{40 a^{3}}+\frac {7 b \,x^{2}}{4 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}-\frac {21 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{8}}\) | \(131\) |
| default | \(-\frac {b^{3} \left (\frac {21 \ln \left (b \,x^{2}+a \right )}{b}-\frac {3 a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}-\frac {2 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}-\frac {5 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {15 a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{8}}-\frac {1}{4 a^{6} x^{4}}+\frac {3 b}{a^{7} x^{2}}+\frac {21 b^{2} \ln \left (x \right )}{a^{8}}\) | \(134\) |
| parallelrisch | \(\frac {-10 a^{7}-4200 \ln \left (b \,x^{2}+a \right ) x^{10} a^{2} b^{5}-4200 \ln \left (b \,x^{2}+a \right ) x^{8} a^{3} b^{4}-2100 \ln \left (b \,x^{2}+a \right ) x^{6} a^{4} b^{3}-420 \ln \left (b \,x^{2}+a \right ) x^{4} a^{5} b^{2}-959 b^{7} x^{14}+8400 \ln \left (x \right ) x^{8} a^{3} b^{4}+4200 \ln \left (x \right ) x^{6} a^{4} b^{3}+840 \ln \left (x \right ) x^{4} a^{5} b^{2}+4200 \ln \left (x \right ) x^{12} a \,b^{6}-2100 \ln \left (b \,x^{2}+a \right ) x^{12} a \,b^{6}+8400 \ln \left (x \right ) x^{10} a^{2} b^{5}-420 \ln \left (b \,x^{2}+a \right ) x^{14} b^{7}-7700 x^{10} a^{2} b^{5}+70 x^{2} a^{6} b -4375 a \,x^{12} b^{6}+840 \ln \left (x \right ) x^{14} b^{7}-6300 x^{8} a^{3} b^{4}-2100 x^{6} a^{4} b^{3}}{40 a^{8} x^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}\) | \(285\) |
Input:
int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
(-1/4/a+7/4*b/a^2*x^2-105/2*b^3/a^4*x^6-315/2*b^4/a^5*x^8-385/2*b^5/a^6*x^ 10-875/8*b^6/a^7*x^12-959/40*b^7/a^8*x^14)/x^4/(b*x^2+a)^5+21*b^2*ln(x)/a^ 8-21/2*b^2*ln(b*x^2+a)/a^8
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (128) = 256\).
Time = 0.08 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {420 \, a b^{6} x^{12} + 1890 \, a^{2} b^{5} x^{10} + 3290 \, a^{3} b^{4} x^{8} + 2695 \, a^{4} b^{3} x^{6} + 959 \, a^{5} b^{2} x^{4} + 70 \, a^{6} b x^{2} - 10 \, a^{7} - 420 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 840 \, {\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (x\right )}{40 \, {\left (a^{8} b^{5} x^{14} + 5 \, a^{9} b^{4} x^{12} + 10 \, a^{10} b^{3} x^{10} + 10 \, a^{11} b^{2} x^{8} + 5 \, a^{12} b x^{6} + a^{13} x^{4}\right )}} \] Input:
integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/40*(420*a*b^6*x^12 + 1890*a^2*b^5*x^10 + 3290*a^3*b^4*x^8 + 2695*a^4*b^3 *x^6 + 959*a^5*b^2*x^4 + 70*a^6*b*x^2 - 10*a^7 - 420*(b^7*x^14 + 5*a*b^6*x ^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*log( b*x^2 + a) + 840*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x ^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*log(x))/(a^8*b^5*x^14 + 5*a^9*b^4*x^12 + 10*a^10*b^3*x^10 + 10*a^11*b^2*x^8 + 5*a^12*b*x^6 + a^13*x^4)
Time = 0.44 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {- 10 a^{6} + 70 a^{5} b x^{2} + 959 a^{4} b^{2} x^{4} + 2695 a^{3} b^{3} x^{6} + 3290 a^{2} b^{4} x^{8} + 1890 a b^{5} x^{10} + 420 b^{6} x^{12}}{40 a^{12} x^{4} + 200 a^{11} b x^{6} + 400 a^{10} b^{2} x^{8} + 400 a^{9} b^{3} x^{10} + 200 a^{8} b^{4} x^{12} + 40 a^{7} b^{5} x^{14}} + \frac {21 b^{2} \log {\left (x \right )}}{a^{8}} - \frac {21 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{8}} \] Input:
integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
(-10*a**6 + 70*a**5*b*x**2 + 959*a**4*b**2*x**4 + 2695*a**3*b**3*x**6 + 32 90*a**2*b**4*x**8 + 1890*a*b**5*x**10 + 420*b**6*x**12)/(40*a**12*x**4 + 2 00*a**11*b*x**6 + 400*a**10*b**2*x**8 + 400*a**9*b**3*x**10 + 200*a**8*b** 4*x**12 + 40*a**7*b**5*x**14) + 21*b**2*log(x)/a**8 - 21*b**2*log(a/b + x* *2)/(2*a**8)
Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {420 \, b^{6} x^{12} + 1890 \, a b^{5} x^{10} + 3290 \, a^{2} b^{4} x^{8} + 2695 \, a^{3} b^{3} x^{6} + 959 \, a^{4} b^{2} x^{4} + 70 \, a^{5} b x^{2} - 10 \, a^{6}}{40 \, {\left (a^{7} b^{5} x^{14} + 5 \, a^{8} b^{4} x^{12} + 10 \, a^{9} b^{3} x^{10} + 10 \, a^{10} b^{2} x^{8} + 5 \, a^{11} b x^{6} + a^{12} x^{4}\right )}} - \frac {21 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{8}} + \frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} \] Input:
integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/40*(420*b^6*x^12 + 1890*a*b^5*x^10 + 3290*a^2*b^4*x^8 + 2695*a^3*b^3*x^6 + 959*a^4*b^2*x^4 + 70*a^5*b*x^2 - 10*a^6)/(a^7*b^5*x^14 + 5*a^8*b^4*x^12 + 10*a^9*b^3*x^10 + 10*a^10*b^2*x^8 + 5*a^11*b*x^6 + a^12*x^4) - 21/2*b^2 *log(b*x^2 + a)/a^8 + 21/2*b^2*log(x^2)/a^8
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} - \frac {21 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{8}} - \frac {63 \, b^{2} x^{4} - 12 \, a b x^{2} + a^{2}}{4 \, a^{8} x^{4}} + \frac {959 \, b^{7} x^{10} + 5095 \, a b^{6} x^{8} + 10890 \, a^{2} b^{5} x^{6} + 11730 \, a^{3} b^{4} x^{4} + 6390 \, a^{4} b^{3} x^{2} + 1418 \, a^{5} b^{2}}{40 \, {\left (b x^{2} + a\right )}^{5} a^{8}} \] Input:
integrate(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
21/2*b^2*log(x^2)/a^8 - 21/2*b^2*log(abs(b*x^2 + a))/a^8 - 1/4*(63*b^2*x^4 - 12*a*b*x^2 + a^2)/(a^8*x^4) + 1/40*(959*b^7*x^10 + 5095*a*b^6*x^8 + 108 90*a^2*b^5*x^6 + 11730*a^3*b^4*x^4 + 6390*a^4*b^3*x^2 + 1418*a^5*b^2)/((b* x^2 + a)^5*a^8)
Time = 18.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {7\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {959\,b^2\,x^4}{40\,a^3}+\frac {539\,b^3\,x^6}{8\,a^4}+\frac {329\,b^4\,x^8}{4\,a^5}+\frac {189\,b^5\,x^{10}}{4\,a^6}+\frac {21\,b^6\,x^{12}}{2\,a^7}}{a^5\,x^4+5\,a^4\,b\,x^6+10\,a^3\,b^2\,x^8+10\,a^2\,b^3\,x^{10}+5\,a\,b^4\,x^{12}+b^5\,x^{14}}-\frac {21\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^8}+\frac {21\,b^2\,\ln \left (x\right )}{a^8} \] Input:
int(1/(x^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)
Output:
((7*b*x^2)/(4*a^2) - 1/(4*a) + (959*b^2*x^4)/(40*a^3) + (539*b^3*x^6)/(8*a ^4) + (329*b^4*x^8)/(4*a^5) + (189*b^5*x^10)/(4*a^6) + (21*b^6*x^12)/(2*a^ 7))/(a^5*x^4 + b^5*x^14 + 5*a^4*b*x^6 + 5*a*b^4*x^12 + 10*a^3*b^2*x^8 + 10 *a^2*b^3*x^10) - (21*b^2*log(a + b*x^2))/(2*a^8) + (21*b^2*log(x))/a^8
Time = 0.17 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {-420 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} b^{2} x^{4}-2100 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b^{3} x^{6}-4200 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{4} x^{8}-4200 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{5} x^{10}-2100 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{6} x^{12}-420 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{7} x^{14}+840 \,\mathrm {log}\left (x \right ) a^{5} b^{2} x^{4}+4200 \,\mathrm {log}\left (x \right ) a^{4} b^{3} x^{6}+8400 \,\mathrm {log}\left (x \right ) a^{3} b^{4} x^{8}+8400 \,\mathrm {log}\left (x \right ) a^{2} b^{5} x^{10}+4200 \,\mathrm {log}\left (x \right ) a \,b^{6} x^{12}+840 \,\mathrm {log}\left (x \right ) b^{7} x^{14}-10 a^{7}+70 a^{6} b \,x^{2}+875 a^{5} b^{2} x^{4}+2275 a^{4} b^{3} x^{6}+2450 a^{3} b^{4} x^{8}+1050 a^{2} b^{5} x^{10}-84 b^{7} x^{14}}{40 a^{8} x^{4} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:
int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
( - 420*log(a + b*x**2)*a**5*b**2*x**4 - 2100*log(a + b*x**2)*a**4*b**3*x* *6 - 4200*log(a + b*x**2)*a**3*b**4*x**8 - 4200*log(a + b*x**2)*a**2*b**5* x**10 - 2100*log(a + b*x**2)*a*b**6*x**12 - 420*log(a + b*x**2)*b**7*x**14 + 840*log(x)*a**5*b**2*x**4 + 4200*log(x)*a**4*b**3*x**6 + 8400*log(x)*a* *3*b**4*x**8 + 8400*log(x)*a**2*b**5*x**10 + 4200*log(x)*a*b**6*x**12 + 84 0*log(x)*b**7*x**14 - 10*a**7 + 70*a**6*b*x**2 + 875*a**5*b**2*x**4 + 2275 *a**4*b**3*x**6 + 2450*a**3*b**4*x**8 + 1050*a**2*b**5*x**10 - 84*b**7*x** 14)/(40*a**8*x**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))