Integrand size = 23, antiderivative size = 161 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {d}{2 \left (b^2 c^2-a^2 d^2\right ) (c+d x)^2}+\frac {2 b^2 c d}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)}-\frac {b^2 \log (a-b x)}{2 a (b c+a d)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3}-\frac {b^2 d \left (3 b^2 c^2+a^2 d^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3} \]
1/2*d/(-a^2*d^2+b^2*c^2)/(d*x+c)^2+2*b^2*c*d/(-a^2*d^2+b^2*c^2)^2/(d*x+c)- 1/2*b^2*ln(-b*x+a)/a/(a*d+b*c)^3+1/2*b^2*ln(b*x+a)/a/(-a*d+b*c)^3-b^2*d*(a ^2*d^2+3*b^2*c^2)*ln(d*x+c)/(-a^2*d^2+b^2*c^2)^3
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {1}{2} \left (-\frac {b^2 \log (a-b x)}{a (b c+a d)^3}-\frac {b^2 \log (a+b x)}{a (-b c+a d)^3}+\frac {d \left (\frac {\left (b^2 c^2-a^2 d^2\right ) \left (-a^2 d^2+b^2 c (5 c+4 d x)\right )}{(c+d x)^2}-2 \left (3 b^4 c^2+a^2 b^2 d^2\right ) \log (c+d x)\right )}{\left (b^2 c^2-a^2 d^2\right )^3}\right ) \]
(-((b^2*Log[a - b*x])/(a*(b*c + a*d)^3)) - (b^2*Log[a + b*x])/(a*(-(b*c) + a*d)^3) + (d*(((b^2*c^2 - a^2*d^2)*(-(a^2*d^2) + b^2*c*(5*c + 4*d*x)))/(c + d*x)^2 - 2*(3*b^4*c^2 + a^2*b^2*d^2)*Log[c + d*x]))/(b^2*c^2 - a^2*d^2) ^3)/2
Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {82, 477, 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.
| \(\displaystyle \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 82 |
| \(\displaystyle \int \frac {1}{\left (a^2-b^2 x^2\right ) (c+d x)^3}dx\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (\frac {a b^3}{2 (b c+a d)^3 (a-b x)}+\frac {a b^3}{2 (b c-a d)^3 (a+b x)}-\frac {a^2 d^2 \left (3 b^2 c^2+a^2 d^2\right ) b^2}{\left (b^2 c^2-a^2 d^2\right )^3 (c+d x)}-\frac {2 a^2 c d^2 b^2}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)^2}-\frac {a^2 d^2}{\left (b^2 c^2-a^2 d^2\right ) (c+d x)^3}\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac {a^2 d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac {a^2 b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac {a b^2 \log (a-b x)}{2 (a d+b c)^3}+\frac {a b^2 \log (a+b x)}{2 (b c-a d)^3}}{a^2}\) |
((a^2*d)/(2*(b^2*c^2 - a^2*d^2)*(c + d*x)^2) + (2*a^2*b^2*c*d)/((b^2*c^2 - a^2*d^2)^2*(c + d*x)) - (a*b^2*Log[a - b*x])/(2*(b*c + a*d)^3) + (a*b^2*L og[a + b*x])/(2*(b*c - a*d)^3) - (a^2*b^2*d*(3*b^2*c^2 + a^2*d^2)*Log[c + d*x])/(b^2*c^2 - a^2*d^2)^3)/a^2
3.16.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Time = 0.97 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {d}{2 \left (a d +b c \right ) \left (a d -b c \right ) \left (d x +c \right )^{2}}+\frac {b^{2} d \left (a^{2} d^{2}+3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d +b c \right )^{3} \left (a d -b c \right )^{3}}+\frac {2 b^{2} d c}{\left (a d +b c \right )^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{2 a \left (a d -b c \right )^{3}}-\frac {b^{2} \ln \left (-b x +a \right )}{2 a \left (a d +b c \right )^{3}}\) | \(158\) |
| norman | \(\frac {\frac {-a^{2} d^{5}+5 b^{2} c^{2} d^{3}}{2 d^{2} \left (a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}\right )}+\frac {2 c \,b^{2} d^{2} x}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}}{\left (d x +c \right )^{2}}+\frac {b^{2} d \left (a^{2} d^{2}+3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}-\frac {b^{2} \ln \left (b x +a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a}-\frac {b^{2} \ln \left (-b x +a \right )}{2 \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) a}\) | \(285\) |
| risch | \(\frac {\frac {2 c \,b^{2} d^{2} x}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}-\frac {\left (a^{2} d^{2}-5 b^{2} c^{2}\right ) d}{2 \left (a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}\right )}}{\left (d x +c \right )^{2}}+\frac {b^{2} d^{3} \ln \left (d x +c \right ) a^{2}}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}+\frac {3 b^{4} d \ln \left (d x +c \right ) c^{2}}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}-\frac {b^{2} \ln \left (b x +a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a}-\frac {b^{2} \ln \left (-b x +a \right )}{2 \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) a}\) | \(329\) |
| parallelrisch | \(-\frac {2 \ln \left (b x -a \right ) x \,a^{3} b^{2} c \,d^{6}-6 \ln \left (b x -a \right ) x \,a^{2} b^{3} c^{2} d^{5}+3 \ln \left (b x +a \right ) a^{2} b^{3} c^{3} d^{4}+3 \ln \left (b x +a \right ) a \,b^{4} c^{4} d^{3}-2 \ln \left (d x +c \right ) a^{3} b^{2} c^{2} d^{5}+6 \ln \left (b x -a \right ) x a \,b^{4} c^{3} d^{4}-6 \ln \left (d x +c \right ) a \,b^{4} c^{4} d^{3}-4 x \,a^{3} b^{2} c \,d^{6}+4 x a \,b^{4} c^{3} d^{4}+\ln \left (b x -a \right ) x^{2} a^{3} b^{2} d^{7}-\ln \left (b x -a \right ) x^{2} b^{5} c^{3} d^{4}+\ln \left (b x +a \right ) x^{2} a^{3} b^{2} d^{7}+\ln \left (b x +a \right ) x^{2} b^{5} c^{3} d^{4}-2 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} d^{7}-2 \ln \left (b x -a \right ) x \,b^{5} c^{4} d^{3}-6 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{2} d^{5}+2 \ln \left (b x +a \right ) x \,b^{5} c^{4} d^{3}+\ln \left (b x -a \right ) a^{3} b^{2} c^{2} d^{5}-3 \ln \left (b x -a \right ) a^{2} b^{3} c^{3} d^{4}+3 \ln \left (b x -a \right ) a \,b^{4} c^{4} d^{3}+\ln \left (b x +a \right ) a^{3} b^{2} c^{2} d^{5}-6 a^{3} b^{2} c^{2} d^{5}+5 a \,b^{4} c^{4} d^{3}-\ln \left (b x -a \right ) b^{5} c^{5} d^{2}+\ln \left (b x +a \right ) b^{5} c^{5} d^{2}+2 \ln \left (b x +a \right ) x \,a^{3} b^{2} c \,d^{6}+6 \ln \left (b x +a \right ) x \,a^{2} b^{3} c^{2} d^{5}+6 \ln \left (b x +a \right ) x a \,b^{4} c^{3} d^{4}-4 \ln \left (d x +c \right ) x \,a^{3} b^{2} c \,d^{6}-12 \ln \left (d x +c \right ) x a \,b^{4} c^{3} d^{4}-3 \ln \left (b x -a \right ) x^{2} a^{2} b^{3} c \,d^{6}+3 \ln \left (b x -a \right ) x^{2} a \,b^{4} c^{2} d^{5}+3 \ln \left (b x +a \right ) x^{2} a^{2} b^{3} c \,d^{6}+3 \ln \left (b x +a \right ) x^{2} a \,b^{4} c^{2} d^{5}+a^{5} d^{7}}{2 \left (a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}\right ) a \left (d x +c \right )^{2} d^{2}}\) | \(725\) |
-1/2*d/(a*d+b*c)/(a*d-b*c)/(d*x+c)^2+b^2*d*(a^2*d^2+3*b^2*c^2)/(a*d+b*c)^3 /(a*d-b*c)^3*ln(d*x+c)+2*b^2*d*c/(a*d+b*c)^2/(a*d-b*c)^2/(d*x+c)-1/2/a*b^2 /(a*d-b*c)^3*ln(b*x+a)-1/2*b^2*ln(-b*x+a)/a/(a*d+b*c)^3
Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (155) = 310\).
Time = 2.49 (sec) , antiderivative size = 605, normalized size of antiderivative = 3.76 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {5 \, a b^{4} c^{4} d - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5} + 4 \, {\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x + {\left (b^{5} c^{5} + 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} + 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d + 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) - {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x - a\right ) - 2 \, {\left (3 \, a b^{4} c^{4} d + a^{3} b^{2} c^{2} d^{3} + {\left (3 \, a b^{4} c^{2} d^{3} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{8} - 3 \, a^{3} b^{4} c^{6} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{4} - a^{7} c^{2} d^{6} + {\left (a b^{6} c^{6} d^{2} - 3 \, a^{3} b^{4} c^{4} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} d - 3 \, a^{3} b^{4} c^{5} d^{3} + 3 \, a^{5} b^{2} c^{3} d^{5} - a^{7} c d^{7}\right )} x\right )}} \]
1/2*(5*a*b^4*c^4*d - 6*a^3*b^2*c^2*d^3 + a^5*d^5 + 4*(a*b^4*c^3*d^2 - a^3* b^2*c*d^4)*x + (b^5*c^5 + 3*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + a^3*b^2*c^2* d^3 + (b^5*c^3*d^2 + 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + 2*(b^5*c^4*d + 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*x)*l og(b*x + a) - (b^5*c^5 - 3*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d ^3 + (b^5*c^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*x^2 + 2*(b^5*c^4*d - 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 - a^3*b^2*c*d^4)*x)*lo g(b*x - a) - 2*(3*a*b^4*c^4*d + a^3*b^2*c^2*d^3 + (3*a*b^4*c^2*d^3 + a^3*b ^2*d^5)*x^2 + 2*(3*a*b^4*c^3*d^2 + a^3*b^2*c*d^4)*x)*log(d*x + c))/(a*b^6* c^8 - 3*a^3*b^4*c^6*d^2 + 3*a^5*b^2*c^4*d^4 - a^7*c^2*d^6 + (a*b^6*c^6*d^2 - 3*a^3*b^4*c^4*d^4 + 3*a^5*b^2*c^2*d^6 - a^7*d^8)*x^2 + 2*(a*b^6*c^7*d - 3*a^3*b^4*c^5*d^3 + 3*a^5*b^2*c^3*d^5 - a^7*c*d^7)*x)
Timed out. \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (155) = 310\).
Time = 0.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.95 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {b^{2} \log \left (b x + a\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} - \frac {b^{2} \log \left (b x - a\right )}{2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d + a^{2} b^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{6} c^{6} - 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{6} d^{6}} + \frac {4 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - a^{2} d^{3}}{2 \, {\left (b^{4} c^{6} - 2 \, a^{2} b^{2} c^{4} d^{2} + a^{4} c^{2} d^{4} + {\left (b^{4} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{4} d^{6}\right )} x^{2} + 2 \, {\left (b^{4} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} \]
1/2*b^2*log(b*x + a)/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^ 3) - 1/2*b^2*log(b*x - a)/(a*b^3*c^3 + 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 + a ^4*d^3) - (3*b^4*c^2*d + a^2*b^2*d^3)*log(d*x + c)/(b^6*c^6 - 3*a^2*b^4*c^ 4*d^2 + 3*a^4*b^2*c^2*d^4 - a^6*d^6) + 1/2*(4*b^2*c*d^2*x + 5*b^2*c^2*d - a^2*d^3)/(b^4*c^6 - 2*a^2*b^2*c^4*d^2 + a^4*c^2*d^4 + (b^4*c^4*d^2 - 2*a^2 *b^2*c^2*d^4 + a^4*d^6)*x^2 + 2*(b^4*c^5*d - 2*a^2*b^2*c^3*d^3 + a^4*c*d^5 )*x)
Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )}} - \frac {b^{3} \log \left ({\left | b x - a \right |}\right )}{2 \, {\left (a b^{4} c^{3} + 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} - a^{6} d^{7}} + \frac {5 \, b^{4} c^{4} d - 6 \, a^{2} b^{2} c^{2} d^{3} + a^{4} d^{5} + 4 \, {\left (b^{4} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x}{2 \, {\left (b c + a d\right )}^{3} {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2}} \]
1/2*b^3*log(abs(b*x + a))/(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3) - 1/2*b^3*log(abs(b*x - a))/(a*b^4*c^3 + 3*a^2*b^3*c^2*d + 3*a ^3*b^2*c*d^2 + a^4*b*d^3) - (3*b^4*c^2*d^2 + a^2*b^2*d^4)*log(abs(d*x + c) )/(b^6*c^6*d - 3*a^2*b^4*c^4*d^3 + 3*a^4*b^2*c^2*d^5 - a^6*d^7) + 1/2*(5*b ^4*c^4*d - 6*a^2*b^2*c^2*d^3 + a^4*d^5 + 4*(b^4*c^3*d^2 - a^2*b^2*c*d^4)*x )/((b*c + a*d)^3*(b*c - a*d)^3*(d*x + c)^2)
Time = 1.99 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx=\frac {\ln \left (c+d\,x\right )\,\left (a^2\,b^2\,d^3+3\,b^4\,c^2\,d\right )}{a^6\,d^6-3\,a^4\,b^2\,c^2\,d^4+3\,a^2\,b^4\,c^4\,d^2-b^6\,c^6}-\frac {b^2\,\ln \left (a+b\,x\right )}{2\,\left (a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3\right )}-\frac {b^2\,\ln \left (a-b\,x\right )}{2\,\left (a^4\,d^3+3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d+a\,b^3\,c^3\right )}-\frac {\frac {a^2\,d^3-5\,b^2\,c^2\,d}{2\,\left (a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4\right )}-\frac {2\,b^2\,c\,d^2\,x}{a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4}}{c^2+2\,c\,d\,x+d^2\,x^2} \]
(log(c + d*x)*(3*b^4*c^2*d + a^2*b^2*d^3))/(a^6*d^6 - b^6*c^6 + 3*a^2*b^4* c^4*d^2 - 3*a^4*b^2*c^2*d^4) - (b^2*log(a + b*x))/(2*(a^4*d^3 - a*b^3*c^3 + 3*a^2*b^2*c^2*d - 3*a^3*b*c*d^2)) - (b^2*log(a - b*x))/(2*(a^4*d^3 + a*b ^3*c^3 + 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)) - ((a^2*d^3 - 5*b^2*c^2*d)/(2*( a^4*d^4 + b^4*c^4 - 2*a^2*b^2*c^2*d^2)) - (2*b^2*c*d^2*x)/(a^4*d^4 + b^4*c ^4 - 2*a^2*b^2*c^2*d^2))/(c^2 + d^2*x^2 + 2*c*d*x)