Integrand size = 24, antiderivative size = 116 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1}{2 a^6 x^2}-\frac {b}{10 a^2 \left (a+b x^2\right )^5}-\frac {b}{4 a^3 \left (a+b x^2\right )^4}-\frac {b}{2 a^4 \left (a+b x^2\right )^3}-\frac {b}{a^5 \left (a+b x^2\right )^2}-\frac {5 b}{2 a^6 \left (a+b x^2\right )}-\frac {6 b \log (x)}{a^7}+\frac {3 b \log \left (a+b x^2\right )}{a^7} \]
-1/2/a^6/x^2-1/10*b/a^2/(b*x^2+a)^5-1/4*b/a^3/(b*x^2+a)^4-1/2*b/a^4/(b*x^2 +a)^3-b/a^5/(b*x^2+a)^2-5/2*b/a^6/(b*x^2+a)-6*b*ln(x)/a^7+3*b*ln(b*x^2+a)/ a^7
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {a \left (10 a^5+137 a^4 b x^2+385 a^3 b^2 x^4+470 a^2 b^3 x^6+270 a b^4 x^8+60 b^5 x^{10}\right )}{x^2 \left (a+b x^2\right )^5}+120 b \log (x)-60 b \log \left (a+b x^2\right )}{20 a^7} \]
-1/20*((a*(10*a^5 + 137*a^4*b*x^2 + 385*a^3*b^2*x^4 + 470*a^2*b^3*x^6 + 27 0*a*b^4*x^8 + 60*b^5*x^10))/(x^2*(a + b*x^2)^5) + 120*b*Log[x] - 60*b*Log[ a + b*x^2])/a^7
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^6 \int \frac {1}{b^6 x^3 \left (b x^2+a\right )^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^6}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^6}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {6 b^2}{a^7 \left (b x^2+a\right )}+\frac {5 b^2}{a^6 \left (b x^2+a\right )^2}+\frac {4 b^2}{a^5 \left (b x^2+a\right )^3}+\frac {3 b^2}{a^4 \left (b x^2+a\right )^4}+\frac {2 b^2}{a^3 \left (b x^2+a\right )^5}+\frac {b^2}{a^2 \left (b x^2+a\right )^6}-\frac {6 b}{a^7 x^2}+\frac {1}{a^6 x^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {6 b \log \left (x^2\right )}{a^7}+\frac {6 b \log \left (a+b x^2\right )}{a^7}-\frac {5 b}{a^6 \left (a+b x^2\right )}-\frac {1}{a^6 x^2}-\frac {2 b}{a^5 \left (a+b x^2\right )^2}-\frac {b}{a^4 \left (a+b x^2\right )^3}-\frac {b}{2 a^3 \left (a+b x^2\right )^4}-\frac {b}{5 a^2 \left (a+b x^2\right )^5}\right )\) |
(-(1/(a^6*x^2)) - b/(5*a^2*(a + b*x^2)^5) - b/(2*a^3*(a + b*x^2)^4) - b/(a ^4*(a + b*x^2)^3) - (2*b)/(a^5*(a + b*x^2)^2) - (5*b)/(a^6*(a + b*x^2)) - (6*b*Log[x^2])/a^7 + (6*b*Log[a + b*x^2])/a^7)/2
3.6.21.3.1 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_))^(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 = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {15 b^{2} x^{4}}{a^{3}}+\frac {45 b^{3} x^{6}}{a^{4}}+\frac {55 b^{4} x^{8}}{a^{5}}+\frac {125 b^{5} x^{10}}{4 a^{6}}+\frac {137 b^{6} x^{12}}{20 a^{7}}}{x^{2} \left (b \,x^{2}+a \right )^{5}}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {3 b \ln \left (b \,x^{2}+a \right )}{a^{7}}\) | \(98\) |
risch | \(\frac {-\frac {3 b^{5} x^{10}}{a^{6}}-\frac {27 b^{4} x^{8}}{2 a^{5}}-\frac {47 b^{3} x^{6}}{2 a^{4}}-\frac {77 b^{2} x^{4}}{4 a^{3}}-\frac {137 b \,x^{2}}{20 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {3 b \ln \left (-b \,x^{2}-a \right )}{a^{7}}\) | \(119\) |
default | \(-\frac {1}{2 a^{6} x^{2}}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {b^{2} \left (-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}+\frac {6 \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {2 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {5 a}{b \left (b \,x^{2}+a \right )}-\frac {a^{4}}{2 b \left (b \,x^{2}+a \right )^{4}}\right )}{2 a^{7}}\) | \(123\) |
parallelrisch | \(-\frac {120 b^{6} \ln \left (x \right ) x^{12}-60 \ln \left (b \,x^{2}+a \right ) x^{12} b^{6}-137 b^{6} x^{12}+600 b^{5} a \ln \left (x \right ) x^{10}-300 \ln \left (b \,x^{2}+a \right ) x^{10} a \,b^{5}-625 a \,b^{5} x^{10}+1200 a^{2} b^{4} \ln \left (x \right ) x^{8}-600 \ln \left (b \,x^{2}+a \right ) x^{8} a^{2} b^{4}-1100 a^{2} b^{4} x^{8}+1200 a^{3} b^{3} \ln \left (x \right ) x^{6}-600 \ln \left (b \,x^{2}+a \right ) x^{6} a^{3} b^{3}-900 a^{3} b^{3} x^{6}+600 a^{4} b^{2} \ln \left (x \right ) x^{4}-300 \ln \left (b \,x^{2}+a \right ) x^{4} a^{4} b^{2}-300 a^{4} b^{2} x^{4}+120 a^{5} b \ln \left (x \right ) x^{2}-60 \ln \left (b \,x^{2}+a \right ) x^{2} a^{5} b +10 a^{6}}{20 a^{7} x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}\) | \(272\) |
(-1/2/a+15*b^2/a^3*x^4+45*b^3/a^4*x^6+55*b^4/a^5*x^8+125/4*b^5/a^6*x^10+13 7/20*b^6/a^7*x^12)/x^2/(b*x^2+a)^5-6*b*ln(x)/a^7+3*b*ln(b*x^2+a)/a^7
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (106) = 212\).
Time = 0.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.16 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {60 \, a b^{5} x^{10} + 270 \, a^{2} b^{4} x^{8} + 470 \, a^{3} b^{3} x^{6} + 385 \, a^{4} b^{2} x^{4} + 137 \, a^{5} b x^{2} + 10 \, a^{6} - 60 \, {\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (x\right )}{20 \, {\left (a^{7} b^{5} x^{12} + 5 \, a^{8} b^{4} x^{10} + 10 \, a^{9} b^{3} x^{8} + 10 \, a^{10} b^{2} x^{6} + 5 \, a^{11} b x^{4} + a^{12} x^{2}\right )}} \]
-1/20*(60*a*b^5*x^10 + 270*a^2*b^4*x^8 + 470*a^3*b^3*x^6 + 385*a^4*b^2*x^4 + 137*a^5*b*x^2 + 10*a^6 - 60*(b^6*x^12 + 5*a*b^5*x^10 + 10*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 5*a^4*b^2*x^4 + a^5*b*x^2)*log(b*x^2 + a) + 120*(b^6*x^1 2 + 5*a*b^5*x^10 + 10*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 5*a^4*b^2*x^4 + a^5*b *x^2)*log(x))/(a^7*b^5*x^12 + 5*a^8*b^4*x^10 + 10*a^9*b^3*x^8 + 10*a^10*b^ 2*x^6 + 5*a^11*b*x^4 + a^12*x^2)
Time = 0.42 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {- 10 a^{5} - 137 a^{4} b x^{2} - 385 a^{3} b^{2} x^{4} - 470 a^{2} b^{3} x^{6} - 270 a b^{4} x^{8} - 60 b^{5} x^{10}}{20 a^{11} x^{2} + 100 a^{10} b x^{4} + 200 a^{9} b^{2} x^{6} + 200 a^{8} b^{3} x^{8} + 100 a^{7} b^{4} x^{10} + 20 a^{6} b^{5} x^{12}} - \frac {6 b \log {\left (x \right )}}{a^{7}} + \frac {3 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{7}} \]
(-10*a**5 - 137*a**4*b*x**2 - 385*a**3*b**2*x**4 - 470*a**2*b**3*x**6 - 27 0*a*b**4*x**8 - 60*b**5*x**10)/(20*a**11*x**2 + 100*a**10*b*x**4 + 200*a** 9*b**2*x**6 + 200*a**8*b**3*x**8 + 100*a**7*b**4*x**10 + 20*a**6*b**5*x**1 2) - 6*b*log(x)/a**7 + 3*b*log(a/b + x**2)/a**7
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {60 \, b^{5} x^{10} + 270 \, a b^{4} x^{8} + 470 \, a^{2} b^{3} x^{6} + 385 \, a^{3} b^{2} x^{4} + 137 \, a^{4} b x^{2} + 10 \, a^{5}}{20 \, {\left (a^{6} b^{5} x^{12} + 5 \, a^{7} b^{4} x^{10} + 10 \, a^{8} b^{3} x^{8} + 10 \, a^{9} b^{2} x^{6} + 5 \, a^{10} b x^{4} + a^{11} x^{2}\right )}} + \frac {3 \, b \log \left (b x^{2} + a\right )}{a^{7}} - \frac {3 \, b \log \left (x^{2}\right )}{a^{7}} \]
-1/20*(60*b^5*x^10 + 270*a*b^4*x^8 + 470*a^2*b^3*x^6 + 385*a^3*b^2*x^4 + 1 37*a^4*b*x^2 + 10*a^5)/(a^6*b^5*x^12 + 5*a^7*b^4*x^10 + 10*a^8*b^3*x^8 + 1 0*a^9*b^2*x^6 + 5*a^10*b*x^4 + a^11*x^2) + 3*b*log(b*x^2 + a)/a^7 - 3*b*lo g(x^2)/a^7
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {3 \, b \log \left (x^{2}\right )}{a^{7}} + \frac {3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{7}} + \frac {6 \, b x^{2} - a}{2 \, a^{7} x^{2}} - \frac {137 \, b^{6} x^{10} + 735 \, a b^{5} x^{8} + 1590 \, a^{2} b^{4} x^{6} + 1740 \, a^{3} b^{3} x^{4} + 970 \, a^{4} b^{2} x^{2} + 224 \, a^{5} b}{20 \, {\left (b x^{2} + a\right )}^{5} a^{7}} \]
-3*b*log(x^2)/a^7 + 3*b*log(abs(b*x^2 + a))/a^7 + 1/2*(6*b*x^2 - a)/(a^7*x ^2) - 1/20*(137*b^6*x^10 + 735*a*b^5*x^8 + 1590*a^2*b^4*x^6 + 1740*a^3*b^3 *x^4 + 970*a^4*b^2*x^2 + 224*a^5*b)/((b*x^2 + a)^5*a^7)
Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {3\,b\,\ln \left (b\,x^2+a\right )}{a^7}-\frac {\frac {1}{2\,a}+\frac {137\,b\,x^2}{20\,a^2}+\frac {77\,b^2\,x^4}{4\,a^3}+\frac {47\,b^3\,x^6}{2\,a^4}+\frac {27\,b^4\,x^8}{2\,a^5}+\frac {3\,b^5\,x^{10}}{a^6}}{a^5\,x^2+5\,a^4\,b\,x^4+10\,a^3\,b^2\,x^6+10\,a^2\,b^3\,x^8+5\,a\,b^4\,x^{10}+b^5\,x^{12}}-\frac {6\,b\,\ln \left (x\right )}{a^7} \]