Integrand size = 22, antiderivative size = 141 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^2}{2 a (b c-a d)^2 \left (a+b x^2\right )}+\frac {d^2}{2 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\log (x)}{a^2 c^2}-\frac {b^2 (b c-3 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}-\frac {d^2 (3 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^3} \]
1/2*b^2/a/(-a*d+b*c)^2/(b*x^2+a)+1/2*d^2/c/(-a*d+b*c)^2/(d*x^2+c)+ln(x)/a^ 2/c^2-1/2*b^2*(-3*a*d+b*c)*ln(b*x^2+a)/a^2/(-a*d+b*c)^3-1/2*d^2*(-a*d+3*b* c)*ln(d*x^2+c)/c^2/(-a*d+b*c)^3
Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {b^2}{a (b c-a d)^2 \left (a+b x^2\right )}+\frac {d^2}{c (b c-a d)^2 \left (c+d x^2\right )}+\frac {2 \log (x)}{a^2 c^2}+\frac {b^2 (b c-3 a d) \log \left (a+b x^2\right )}{a^2 (-b c+a d)^3}+\frac {d^2 (-3 b c+a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^3}\right ) \]
(b^2/(a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)) + ( 2*Log[x])/(a^2*c^2) + (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(a^2*(-(b*c) + a* d)^3) + (d^2*(-3*b*c + a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^3))/2
Time = 0.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 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.
| \(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^2}dx^2\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {(3 a d-b c) b^3}{a^2 (a d-b c)^3 \left (b x^2+a\right )}-\frac {b^3}{a (a d-b c)^2 \left (b x^2+a\right )^2}-\frac {d^3 (3 b c-a d)}{c^2 (b c-a d)^3 \left (d x^2+c\right )}+\frac {1}{a^2 c^2 x^2}-\frac {d^3}{c (b c-a d)^2 \left (d x^2+c\right )^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b^2 (b c-3 a d) \log \left (a+b x^2\right )}{a^2 (b c-a d)^3}+\frac {\log \left (x^2\right )}{a^2 c^2}+\frac {b^2}{a \left (a+b x^2\right ) (b c-a d)^2}-\frac {d^2 (3 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^3}+\frac {d^2}{c \left (c+d x^2\right ) (b c-a d)^2}\right )\) |
(b^2/(a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)) + L og[x^2]/(a^2*c^2) - (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(a^2*(b*c - a*d)^3) - (d^2*(3*b*c - a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^3))/2
3.4.6.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.77 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{a^{2} c^{2}}-\frac {b^{3} \left (\frac {\left (3 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{2} \left (a d -b c \right )^{3}}-\frac {d^{3} \left (\frac {\left (a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 c^{2} \left (a d -b c \right )^{3}}\) | \(136\) |
| norman | \(\frac {\frac {\left (-a^{3} d^{3}-b^{3} c^{3}\right ) x^{2}}{2 c^{2} a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-a^{2} d^{2}-b^{2} c^{2}\right ) b d \,x^{4}}{2 c^{2} a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {\ln \left (x \right )}{a^{2} c^{2}}-\frac {b^{2} \left (3 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{2} \left (a d -3 b c \right ) \ln \left (d \,x^{2}+c \right )}{2 c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(260\) |
| risch | \(\frac {\frac {\left (a d +b c \right ) b d \,x^{2}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a^{2} d^{2}+b^{2} c^{2}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {\ln \left (x \right )}{a^{2} c^{2}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right ) d}{2 a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{3} \ln \left (b \,x^{2}+a \right ) c}{2 a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{3} \ln \left (-d \,x^{2}-c \right ) a}{2 c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d \,x^{2}-c \right ) b}{2 c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(346\) |
| parallelrisch | \(\frac {x^{4} a^{2} b^{2} c \,d^{3}-x^{4} a \,b^{3} c^{2} d^{2}+x^{2} a^{3} b c \,d^{3}-x^{2} a \,b^{3} c^{3} d +2 \ln \left (x \right ) x^{4} a^{3} b \,d^{4}-2 \ln \left (x \right ) x^{4} b^{4} c^{3} d +\ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{3} d -\ln \left (d \,x^{2}+c \right ) x^{4} a^{3} b \,d^{4}-6 \ln \left (x \right ) a^{3} b \,c^{2} d^{2}+6 \ln \left (x \right ) a^{2} b^{2} c^{3} d -3 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{3} d +3 \ln \left (d \,x^{2}+c \right ) a^{3} b \,c^{2} d^{2}-x^{2} a^{4} d^{4}+3 \ln \left (d \,x^{2}+c \right ) x^{4} a^{2} b^{2} c \,d^{3}-4 \ln \left (x \right ) x^{2} a^{3} b c \,d^{3}+2 \ln \left (x \right ) x^{2} a^{4} d^{4}-2 \ln \left (x \right ) x^{2} b^{4} c^{4}+x^{2} b^{4} c^{4}-3 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} c^{2} d^{2}+4 \ln \left (x \right ) x^{2} a \,b^{3} c^{3} d -3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c^{2} d^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{3} d +2 \ln \left (d \,x^{2}+c \right ) x^{2} a^{3} b c \,d^{3}+3 \ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b^{2} c^{2} d^{2}-6 \ln \left (x \right ) x^{4} a^{2} b^{2} c \,d^{3}+6 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d^{2}-x^{4} a^{3} b \,d^{4}+x^{4} b^{4} c^{3} d +\ln \left (b \,x^{2}+a \right ) x^{2} b^{4} c^{4}-\ln \left (d \,x^{2}+c \right ) x^{2} a^{4} d^{4}+2 \ln \left (x \right ) a^{4} c \,d^{3}-2 \ln \left (x \right ) a \,b^{3} c^{4}+\ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{4}-\ln \left (d \,x^{2}+c \right ) a^{4} c \,d^{3}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) a^{2} c^{2}}\) | \(619\) |
ln(x)/a^2/c^2-1/2*b^3/a^2/(a*d-b*c)^3*((3*a*d-b*c)/b*ln(b*x^2+a)-(a*d-b*c) *a/b/(b*x^2+a))-1/2*d^3/c^2/(a*d-b*c)^3*((a*d-3*b*c)/d*ln(d*x^2+c)-(a*d-b* c)*c/d/(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (133) = 266\).
Time = 2.25 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.83 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x^{2} - {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} + {\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{4} + {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x^{2}\right )}} \]
1/2*(a*b^3*c^4 - a^2*b^2*c^3*d + a^3*b*c^2*d^2 - a^4*c*d^3 + (a*b^3*c^3*d - a^3*b*c*d^3)*x^2 - (a*b^3*c^4 - 3*a^2*b^2*c^3*d + (b^4*c^3*d - 3*a*b^3*c ^2*d^2)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2)*x^2)*log(b*x^2 + a) - (3*a^3*b*c^2*d^2 - a^4*c*d^3 + (3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (3*a^2*b^2*c^2*d^2 + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)*log(d*x^2 + c) + 2*(a* b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a *b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)*log(x))/(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3 + (a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4 *b^2*c^3*d^3 - a^5*b*c^2*d^4)*x^4 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5 *b*c^3*d^3 - a^6*c^2*d^4)*x^2)
Timed out. \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (133) = 266\).
Time = 0.20 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.09 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} + \frac {b^{2} c^{2} + a^{2} d^{2} + {\left (b^{2} c d + a b d^{2}\right )} x^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c^{2}} \]
-1/2*(b^3*c - 3*a*b^2*d)*log(b*x^2 + a)/(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3 *a^4*b*c*d^2 - a^5*d^3) - 1/2*(3*b*c*d^2 - a*d^3)*log(d*x^2 + c)/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) + 1/2*(b^2*c^2 + a^2*d^2 + (b^2*c*d + a*b*d^2)*x^2)/(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a *b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c ^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2) + 1/2*log(x^2)/(a^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (133) = 266\).
Time = 0.32 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.28 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (b^{4} c - 3 \, a b^{3} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )}} - \frac {{\left (3 \, b c d^{3} - a d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )}} + \frac {b^{3} c^{2} d x^{4} - 2 \, a b^{2} c d^{2} x^{4} + a^{2} b d^{3} x^{4} + b^{3} c^{3} x^{2} + a b^{2} c^{2} d x^{2} + a^{2} b c d^{2} x^{2} + a^{3} d^{3} x^{2} + 3 \, a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}}{4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c^{2}} \]
-1/2*(b^4*c - 3*a*b^3*d)*log(abs(b*x^2 + a))/(a^2*b^4*c^3 - 3*a^3*b^3*c^2* d + 3*a^4*b^2*c*d^2 - a^5*b*d^3) - 1/2*(3*b*c*d^3 - a*d^4)*log(abs(d*x^2 + c))/(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4) + 1/4*( b^3*c^2*d*x^4 - 2*a*b^2*c*d^2*x^4 + a^2*b*d^3*x^4 + b^3*c^3*x^2 + a*b^2*c^ 2*d*x^2 + a^2*b*c*d^2*x^2 + a^3*d^3*x^2 + 3*a*b^2*c^3 - 2*a^2*b*c^2*d + 3* a^3*c*d^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)) + 1/2*log(x^2)/(a^2*c^2)
Time = 6.80 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {a^2\,d^2+b^2\,c^2}{2\,a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b\,d\,x^2\,\left (a\,d+b\,c\right )}{2\,a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c}+\frac {\ln \left (x\right )}{a^2\,c^2}-\frac {b^2\,\ln \left (b\,x^2+a\right )\,\left (3\,a\,d-b\,c\right )}{2\,a^2\,{\left (a\,d-b\,c\right )}^3}-\frac {d^2\,\ln \left (d\,x^2+c\right )\,\left (a\,d-3\,b\,c\right )}{2\,c^2\,{\left (a\,d-b\,c\right )}^3} \]
((a^2*d^2 + b^2*c^2)/(2*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^2*(a *d + b*c))/(2*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(a*c + x^2*(a*d + b*c) + b*d*x^4) + log(x)/(a^2*c^2) - (b^2*log(a + b*x^2)*(3*a*d - b*c))/(2*a^2 *(a*d - b*c)^3) - (d^2*log(c + d*x^2)*(a*d - 3*b*c))/(2*c^2*(a*d - b*c)^3)