Integrand size = 22, antiderivative size = 210 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {1}{6 a^2 c x^6}+\frac {2 b c+a d}{4 a^3 c^2 x^4}-\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{2 a^4 c^3 x^2}-\frac {b^4}{2 a^4 (b c-a d) \left (a+b x^2\right )}-\frac {\left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]
-1/6/a^2/c/x^6+1/4*(a*d+2*b*c)/a^3/c^2/x^4+1/2*(-a^2*d^2-2*a*b*c*d-3*b^2*c ^2)/a^4/c^3/x^2-1/2*b^4/a^4/(-a*d+b*c)/(b*x^2+a)-(a^3*d^3+2*a^2*b*c*d^2+3* a*b^2*c^2*d+4*b^3*c^3)*ln(x)/a^5/c^4+1/2*b^4*(-5*a*d+4*b*c)*ln(b*x^2+a)/a^ 5/(-a*d+b*c)^2+1/2*d^5*ln(d*x^2+c)/c^4/(-a*d+b*c)^2
Time = 0.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {1}{12} \left (-\frac {2}{a^2 c x^6}+\frac {6 b c+3 a d}{a^3 c^2 x^4}-\frac {6 \left (3 b^2 c^2+2 a b c d+a^2 d^2\right )}{a^4 c^3 x^2}+\frac {6 b^4}{a^4 (-b c+a d) \left (a+b x^2\right )}-\frac {12 \left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {6 b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{a^5 (b c-a d)^2}+\frac {6 d^5 \log \left (c+d x^2\right )}{c^4 (b c-a d)^2}\right ) \]
(-2/(a^2*c*x^6) + (6*b*c + 3*a*d)/(a^3*c^2*x^4) - (6*(3*b^2*c^2 + 2*a*b*c* d + a^2*d^2))/(a^4*c^3*x^2) + (6*b^4)/(a^4*(-(b*c) + a*d)*(a + b*x^2)) - ( 12*(4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(a^5*c^4) + (6*b^4*(4*b*c - 5*a*d)*Log[a + b*x^2])/(a^5*(b*c - a*d)^2) + (6*d^5*Log [c + d*x^2])/(c^4*(b*c - a*d)^2))/12
Time = 0.43 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (b x^2+a\right )^2 \left (d x^2+c\right )}dx^2\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {d^6}{c^4 (b c-a d)^2 \left (d x^2+c\right )}-\frac {b^5 (5 a d-4 b c)}{a^5 (a d-b c)^2 \left (b x^2+a\right )}+\frac {-4 b^3 c^3-3 a b^2 d c^2-2 a^2 b d^2 c-a^3 d^3}{a^5 c^4 x^2}-\frac {b^5}{a^4 (a d-b c) \left (b x^2+a\right )^2}+\frac {3 b^2 c^2+2 a b d c+a^2 d^2}{a^4 c^3 x^4}+\frac {-2 b c-a d}{a^3 c^2 x^6}+\frac {1}{a^2 c x^8}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{a^5 (b c-a d)^2}-\frac {b^4}{a^4 \left (a+b x^2\right ) (b c-a d)}+\frac {a d+2 b c}{2 a^3 c^2 x^4}-\frac {1}{3 a^2 c x^6}-\frac {a^2 d^2+2 a b c d+3 b^2 c^2}{a^4 c^3 x^2}-\frac {\log \left (x^2\right ) \left (a^3 d^3+2 a^2 b c d^2+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}+\frac {d^5 \log \left (c+d x^2\right )}{c^4 (b c-a d)^2}\right )\) |
(-1/3*1/(a^2*c*x^6) + (2*b*c + a*d)/(2*a^3*c^2*x^4) - (3*b^2*c^2 + 2*a*b*c *d + a^2*d^2)/(a^4*c^3*x^2) - b^4/(a^4*(b*c - a*d)*(a + b*x^2)) - ((4*b^3* c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*Log[x^2])/(a^5*c^4) + (b^4* (4*b*c - 5*a*d)*Log[a + b*x^2])/(a^5*(b*c - a*d)^2) + (d^5*Log[c + d*x^2]) /(c^4*(b*c - a*d)^2))/2
3.3.100.3.1 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 = 2.67 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {1}{6 a^{2} c \,x^{6}}-\frac {-a d -2 b c}{4 x^{4} a^{3} c^{2}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{2 a^{4} c^{3} x^{2}}+\frac {\left (-a^{3} d^{3}-2 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}-\frac {b^{5} \left (\frac {\left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5} \left (a d -b c \right )^{2}}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a d -b c \right )^{2}}\) | \(201\) |
| norman | \(\frac {-\frac {1}{6 a c}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 c^{3} a^{3}}+\frac {\left (a^{3} b \,d^{3}+a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d -4 b^{4} c^{3}\right ) b \,x^{8}}{2 c^{3} \left (a d -b c \right ) a^{5}}}{x^{6} \left (b \,x^{2}+a \right )}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}-\frac {b^{4} \left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(270\) |
| risch | \(\frac {-\frac {b \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{6}}{2 a^{4} c^{3} \left (a d -b c \right )}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 c^{3} a^{3}}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {1}{6 a c}}{x^{6} \left (b \,x^{2}+a \right )}-\frac {\ln \left (x \right ) d^{3}}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b \,d^{2}}{a^{3} c^{3}}-\frac {3 \ln \left (x \right ) b^{2} d}{a^{4} c^{2}}-\frac {4 \ln \left (x \right ) b^{3}}{a^{5} c}+\frac {d^{5} \ln \left (-d \,x^{2}-c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(310\) |
| parallelrisch | \(-\frac {2 a^{6} b \,c^{3} d^{2}-4 a^{5} b^{2} c^{4} d +24 x^{6} a \,b^{6} c^{5}+12 x^{4} a^{2} b^{5} c^{5}-4 x^{2} a^{3} b^{4} c^{5}+48 \ln \left (x \right ) x^{8} b^{7} c^{5}-24 \ln \left (b \,x^{2}+a \right ) x^{8} b^{7} c^{5}-60 \ln \left (x \right ) x^{8} a \,b^{6} c^{4} d +30 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{6} c^{4} d -60 \ln \left (x \right ) x^{6} a^{2} b^{5} c^{4} d +30 \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{5} c^{4} d +2 a^{4} b^{3} c^{5}+2 x^{2} a^{5} b^{2} c^{3} d^{2}+5 x^{2} a^{4} b^{3} c^{4} d +12 \ln \left (x \right ) x^{8} a^{5} b^{2} d^{5}-6 \ln \left (d \,x^{2}+c \right ) x^{8} a^{5} b^{2} d^{5}+12 \ln \left (x \right ) x^{6} a^{6} b \,d^{5}+48 \ln \left (x \right ) x^{6} a \,b^{6} c^{5}-24 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{6} c^{5}-6 \ln \left (d \,x^{2}+c \right ) x^{6} a^{6} b \,d^{5}+6 x^{6} a^{5} b^{2} c \,d^{4}-30 x^{6} a^{2} b^{5} c^{4} d +6 x^{4} a^{6} b c \,d^{4}-3 x^{4} a^{5} b^{2} c^{2} d^{3}-15 x^{4} a^{3} b^{4} c^{4} d -3 x^{2} a^{6} b \,c^{2} d^{3}}{12 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) x^{6} c^{4} b \,a^{5}}\) | \(459\) |
-1/6/a^2/c/x^6-1/4*(-a*d-2*b*c)/x^4/a^3/c^2-1/2*(a^2*d^2+2*a*b*c*d+3*b^2*c ^2)/a^4/c^3/x^2+1/a^5/c^4*(-a^3*d^3-2*a^2*b*c*d^2-3*a*b^2*c^2*d-4*b^3*c^3) *ln(x)-1/2*b^5/a^5/(a*d-b*c)^2*((5*a*d-4*b*c)/b*ln(b*x^2+a)-(a*d-b*c)*a/b/ (b*x^2+a))+1/2*d^5/c^4/(a*d-b*c)^2*ln(d*x^2+c)
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (198) = 396\).
Time = 5.02 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {2 \, a^{4} b^{2} c^{5} - 4 \, a^{5} b c^{4} d + 2 \, a^{6} c^{3} d^{2} + 6 \, {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{5} b c d^{4}\right )} x^{6} + 3 \, {\left (4 \, a^{2} b^{4} c^{5} - 5 \, a^{3} b^{3} c^{4} d - a^{5} b c^{2} d^{3} + 2 \, a^{6} c d^{4}\right )} x^{4} - {\left (4 \, a^{3} b^{3} c^{5} - 5 \, a^{4} b^{2} c^{4} d - 2 \, a^{5} b c^{3} d^{2} + 3 \, a^{6} c^{2} d^{3}\right )} x^{2} - 6 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (a^{5} b d^{5} x^{8} + a^{6} d^{5} x^{6}\right )} \log \left (d x^{2} + c\right ) + 12 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d + a^{5} b d^{5}\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{6} d^{5}\right )} x^{6}\right )} \log \left (x\right )}{12 \, {\left ({\left (a^{5} b^{3} c^{6} - 2 \, a^{6} b^{2} c^{5} d + a^{7} b c^{4} d^{2}\right )} x^{8} + {\left (a^{6} b^{2} c^{6} - 2 \, a^{7} b c^{5} d + a^{8} c^{4} d^{2}\right )} x^{6}\right )}} \]
-1/12*(2*a^4*b^2*c^5 - 4*a^5*b*c^4*d + 2*a^6*c^3*d^2 + 6*(4*a*b^5*c^5 - 5* a^2*b^4*c^4*d + a^5*b*c*d^4)*x^6 + 3*(4*a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - a^ 5*b*c^2*d^3 + 2*a^6*c*d^4)*x^4 - (4*a^3*b^3*c^5 - 5*a^4*b^2*c^4*d - 2*a^5* b*c^3*d^2 + 3*a^6*c^2*d^3)*x^2 - 6*((4*b^6*c^5 - 5*a*b^5*c^4*d)*x^8 + (4*a *b^5*c^5 - 5*a^2*b^4*c^4*d)*x^6)*log(b*x^2 + a) - 6*(a^5*b*d^5*x^8 + a^6*d ^5*x^6)*log(d*x^2 + c) + 12*((4*b^6*c^5 - 5*a*b^5*c^4*d + a^5*b*d^5)*x^8 + (4*a*b^5*c^5 - 5*a^2*b^4*c^4*d + a^6*d^5)*x^6)*log(x))/((a^5*b^3*c^6 - 2* a^6*b^2*c^5*d + a^7*b*c^4*d^2)*x^8 + (a^6*b^2*c^6 - 2*a^7*b*c^5*d + a^8*c^ 4*d^2)*x^6)
Timed out. \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.61 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {d^{5} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )}} + \frac {{\left (4 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )}} - \frac {2 \, a^{3} b c^{3} - 2 \, a^{4} c^{2} d + 6 \, {\left (4 \, b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{6} + 3 \, {\left (4 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{4} - {\left (4 \, a^{2} b^{2} c^{3} - a^{3} b c^{2} d - 3 \, a^{4} c d^{2}\right )} x^{2}}{12 \, {\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{8} + {\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{6}\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} \]
1/2*d^5*log(d*x^2 + c)/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2) + 1/2*(4*b^5* c - 5*a*b^4*d)*log(b*x^2 + a)/(a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2) - 1/12 *(2*a^3*b*c^3 - 2*a^4*c^2*d + 6*(4*b^4*c^3 - a*b^3*c^2*d - a^2*b^2*c*d^2 - a^3*b*d^3)*x^6 + 3*(4*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 - 2*a^4*d^3 )*x^4 - (4*a^2*b^2*c^3 - a^3*b*c^2*d - 3*a^4*c*d^2)*x^2)/((a^4*b^2*c^4 - a ^5*b*c^3*d)*x^8 + (a^5*b*c^4 - a^6*c^3*d)*x^6) - 1/2*(4*b^3*c^3 + 3*a*b^2* c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(x^2)/(a^5*c^4)
Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {d^{6} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )}} + \frac {{\left (4 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{5} b^{3} c^{2} - 2 \, a^{6} b^{2} c d + a^{7} b d^{2}\right )}} - \frac {4 \, b^{6} c x^{2} - 5 \, a b^{5} d x^{2} + 5 \, a b^{5} c - 6 \, a^{2} b^{4} d}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} + \frac {44 \, b^{3} c^{3} x^{6} + 33 \, a b^{2} c^{2} d x^{6} + 22 \, a^{2} b c d^{2} x^{6} + 11 \, a^{3} d^{3} x^{6} - 18 \, a b^{2} c^{3} x^{4} - 12 \, a^{2} b c^{2} d x^{4} - 6 \, a^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{3} x^{2} + 3 \, a^{3} c^{2} d x^{2} - 2 \, a^{3} c^{3}}{12 \, a^{5} c^{4} x^{6}} \]
1/2*d^6*log(abs(d*x^2 + c))/(b^2*c^6*d - 2*a*b*c^5*d^2 + a^2*c^4*d^3) + 1/ 2*(4*b^6*c - 5*a*b^5*d)*log(abs(b*x^2 + a))/(a^5*b^3*c^2 - 2*a^6*b^2*c*d + a^7*b*d^2) - 1/2*(4*b^6*c*x^2 - 5*a*b^5*d*x^2 + 5*a*b^5*c - 6*a^2*b^4*d)/ ((a^5*b^2*c^2 - 2*a^6*b*c*d + a^7*d^2)*(b*x^2 + a)) - 1/2*(4*b^3*c^3 + 3*a *b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*log(x^2)/(a^5*c^4) + 1/12*(44*b^3*c^ 3*x^6 + 33*a*b^2*c^2*d*x^6 + 22*a^2*b*c*d^2*x^6 + 11*a^3*d^3*x^6 - 18*a*b^ 2*c^3*x^4 - 12*a^2*b*c^2*d*x^4 - 6*a^3*c*d^2*x^4 + 6*a^2*b*c^3*x^2 + 3*a^3 *c^2*d*x^2 - 2*a^3*c^3)/(a^5*c^4*x^6)
Time = 6.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (4\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^2-4\,a^6\,b\,c\,d+2\,a^5\,b^2\,c^2}-\frac {\frac {1}{6\,a\,c}-\frac {x^2\,\left (3\,a\,d+4\,b\,c\right )}{12\,a^2\,c^2}+\frac {x^4\,\left (2\,a^2\,d^2+3\,a\,b\,c\,d+4\,b^2\,c^2\right )}{4\,a^3\,c^3}+\frac {x^6\,\left (a^3\,b\,d^3+a^2\,b^2\,c\,d^2+a\,b^3\,c^2\,d-4\,b^4\,c^3\right )}{2\,a^4\,c^3\,\left (a\,d-b\,c\right )}}{b\,x^8+a\,x^6}+\frac {d^5\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^4\,d^2-2\,a\,b\,c^5\,d+b^2\,c^6\right )}-\frac {\ln \left (x\right )\,\left (a^3\,d^3+2\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+4\,b^3\,c^3\right )}{a^5\,c^4} \]
(log(a + b*x^2)*(4*b^5*c - 5*a*b^4*d))/(2*a^7*d^2 + 2*a^5*b^2*c^2 - 4*a^6* b*c*d) - (1/(6*a*c) - (x^2*(3*a*d + 4*b*c))/(12*a^2*c^2) + (x^4*(2*a^2*d^2 + 4*b^2*c^2 + 3*a*b*c*d))/(4*a^3*c^3) + (x^6*(a^3*b*d^3 - 4*b^4*c^3 + a^2 *b^2*c*d^2 + a*b^3*c^2*d))/(2*a^4*c^3*(a*d - b*c)))/(a*x^6 + b*x^8) + (d^5 *log(c + d*x^2))/(2*(b^2*c^6 + a^2*c^4*d^2 - 2*a*b*c^5*d)) - (log(x)*(a^3* d^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2))/(a^5*c^4)