Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {b^2}{(b c-a d)^3 (a+b x)}-\frac {d}{2 (b c-a d)^2 (c+d x)^2}-\frac {2 b d}{(b c-a d)^3 (c+d x)}-\frac {3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac {3 b^2 d \log (c+d x)}{(b c-a d)^4} \]
-b^2/(-a*d+b*c)^3/(b*x+a)-1/2*d/(-a*d+b*c)^2/(d*x+c)^2-2*b*d/(-a*d+b*c)^3/ (d*x+c)-3*b^2*d*ln(b*x+a)/(-a*d+b*c)^4+3*b^2*d*ln(d*x+c)/(-a*d+b*c)^4
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {\frac {2 b^2 (b c-a d)}{a+b x}+\frac {d (b c-a d)^2}{(c+d x)^2}+\frac {4 b d (b c-a d)}{c+d x}+6 b^2 d \log (a+b x)-6 b^2 d \log (c+d x)}{2 (b c-a d)^4} \]
-1/2*((2*b^2*(b*c - a*d))/(a + b*x) + (d*(b*c - a*d)^2)/(c + d*x)^2 + (4*b *d*(b*c - a*d))/(c + d*x) + 6*b^2*d*Log[a + b*x] - 6*b^2*d*Log[c + d*x])/( b*c - a*d)^4
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (-\frac {3 b^3 d}{(a+b x) (b c-a d)^4}+\frac {b^3}{(a+b x)^2 (b c-a d)^3}+\frac {3 b^2 d^2}{(c+d x) (b c-a d)^4}+\frac {2 b d^2}{(c+d x)^2 (b c-a d)^3}+\frac {d^2}{(c+d x)^3 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^2}{(a+b x) (b c-a d)^3}-\frac {3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac {3 b^2 d \log (c+d x)}{(b c-a d)^4}-\frac {2 b d}{(c+d x) (b c-a d)^3}-\frac {d}{2 (c+d x)^2 (b c-a d)^2}\) |
-(b^2/((b*c - a*d)^3*(a + b*x))) - d/(2*(b*c - a*d)^2*(c + d*x)^2) - (2*b* d)/((b*c - a*d)^3*(c + d*x)) - (3*b^2*d*Log[a + b*x])/(b*c - a*d)^4 + (3*b ^2*d*Log[c + d*x])/(b*c - a*d)^4
3.3.97.3.1 Defintions of rubi rules used
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])
Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {d}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {3 d \,b^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {2 d b}{\left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {b^{2}}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {3 d \,b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\) | \(108\) |
| risch | \(\frac {\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {3 \left (a d +3 b c \right ) b d x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {a^{2} d^{2}-5 a b c d -2 b^{2} c^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {3 b^{2} d \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 b^{2} d \ln \left (-d x -c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(309\) |
| norman | \(\frac {\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {-a^{2} b \,d^{4}+5 a \,b^{2} c \,d^{3}+2 b^{3} c^{2} d^{2}}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (3 a \,b^{2} d^{4}+9 b^{3} c \,d^{3}\right ) x}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {3 b^{2} d \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 b^{2} d \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(336\) |
| parallelrisch | \(-\frac {-12 \ln \left (d x +c \right ) x a \,b^{3} c \,d^{4}+12 \ln \left (b x +a \right ) x a \,b^{3} c \,d^{4}-6 x a \,b^{3} c \,d^{4}+6 \ln \left (b x +a \right ) x^{2} a \,b^{3} d^{5}+12 \ln \left (b x +a \right ) x^{2} b^{4} c \,d^{4}-6 \ln \left (d x +c \right ) x^{2} a \,b^{3} d^{5}-12 \ln \left (d x +c \right ) x^{2} b^{4} c \,d^{4}+6 \ln \left (b x +a \right ) x \,b^{4} c^{2} d^{3}-6 \ln \left (d x +c \right ) x \,b^{4} c^{2} d^{3}+6 \ln \left (b x +a \right ) a \,b^{3} c^{2} d^{3}-6 \ln \left (d x +c \right ) a \,b^{3} c^{2} d^{3}+3 a \,c^{2} b^{3} d^{3}-6 x^{2} a \,b^{3} d^{5}+6 x^{2} b^{4} c \,d^{4}-3 x \,a^{2} b^{2} d^{5}+9 x \,b^{4} c^{2} d^{3}+6 \ln \left (b x +a \right ) x^{3} b^{4} d^{5}-6 \ln \left (d x +c \right ) x^{3} b^{4} d^{5}-6 a^{2} b^{2} c \,d^{4}+a^{3} b \,d^{5}+2 b^{4} c^{3} d^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b \,d^{2}}\) | \(389\) |
-1/2*d/(a*d-b*c)^2/(d*x+c)^2+3*d/(a*d-b*c)^4*b^2*ln(d*x+c)+2*d/(a*d-b*c)^3 *b/(d*x+c)+1/(a*d-b*c)^3*b^2/(b*x+a)-3*d/(a*d-b*c)^4*b^2*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (108) = 216\).
Time = 0.23 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.50 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]
-1/2*(2*b^3*c^3 + 3*a*b^2*c^2*d - 6*a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 3*(3*b^3*c^2*d - 2*a*b^2*c*d^2 - a^2*b*d^3)*x + 6*(b^3*d ^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^ 2*c*d^2)*x)*log(b*x + a) - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a *b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(d*x + c))/(a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5 *c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d ^6)*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c ^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b ^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (97) = 194\).
Time = 0.89 (sec) , antiderivative size = 632, normalized size of antiderivative = 5.75 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 b^{2} d \log {\left (x + \frac {- \frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} - \frac {3 b^{2} d \log {\left (x + \frac {\frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} + \frac {- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (3 a b d^{2} + 9 b^{2} c d\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \cdot \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \cdot \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]
3*b**2*d*log(x + (-3*a**5*b**2*d**6/(a*d - b*c)**4 + 15*a**4*b**3*c*d**5/( a*d - b*c)**4 - 30*a**3*b**4*c**2*d**4/(a*d - b*c)**4 + 30*a**2*b**5*c**3* d**3/(a*d - b*c)**4 - 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a*b**2*d**2 + 3*b**7*c**5*d/(a*d - b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(a*d - b*c)**4 - 3*b**2*d*log(x + (3*a**5*b**2*d**6/(a*d - b*c)**4 - 15*a**4*b**3*c*d**5/ (a*d - b*c)**4 + 30*a**3*b**4*c**2*d**4/(a*d - b*c)**4 - 30*a**2*b**5*c**3 *d**3/(a*d - b*c)**4 + 15*a*b**6*c**4*d**2/(a*d - b*c)**4 + 3*a*b**2*d**2 - 3*b**7*c**5*d/(a*d - b*c)**4 + 3*b**3*c*d)/(6*b**3*d**2))/(a*d - b*c)**4 + (-a**2*d**2 + 5*a*b*c*d + 2*b**2*c**2 + 6*b**2*d**2*x**2 + x*(3*a*b*d** 2 + 9*b**2*c*d))/(2*a**4*c**2*d**3 - 6*a**3*b*c**3*d**2 + 6*a**2*b**2*c**4 *d - 2*a*b**3*c**5 + x**3*(2*a**3*b*d**5 - 6*a**2*b**2*c*d**4 + 6*a*b**3*c **2*d**3 - 2*b**4*c**3*d**2) + x**2*(2*a**4*d**5 - 2*a**3*b*c*d**4 - 6*a** 2*b**2*c**2*d**3 + 10*a*b**3*c**3*d**2 - 4*b**4*c**4*d) + x*(4*a**4*c*d**4 - 10*a**3*b*c**2*d**3 + 6*a**2*b**2*c**3*d**2 + 2*a*b**3*c**4*d - 2*b**4* c**5))
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (108) = 216\).
Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.51 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {3 \, b^{2} d \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {3 \, b^{2} d \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
-3*b^2*d*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3 *b*c*d^3 + a^4*d^4) + 3*b^2*d*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^ 2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/2*(6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/(a*b^3*c^5 - 3*a^2*b^2* c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3 *a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b ^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b ^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (108) = 216\).
Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 \, b^{3} d \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {5 \, b^{2} d^{3} + \frac {6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]
3*b^3*d*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4 - 4*a*b^4*c^3 *d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - b^5/((b^6*c^3 - 3* a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b*x + a)) + 1/2*(5*b^2*d^3 + 6*(b^4*c*d^2 - a*b^3*d^3)/((b*x + a)*b))/((b*c - a*d)^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2)
Time = 0.20 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.99 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {-a^2\,d^2+5\,a\,b\,c\,d+2\,b^2\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b\,x\,\left (a\,d^2+3\,b\,c\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}-\frac {6\,b^2\,d\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4} \]
((2*b^2*c^2 - a^2*d^2 + 5*a*b*c*d)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (3*b*x*(a*d^2 + 3*b*c*d))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b ^2*c^2*d - 3*a^2*b*c*d^2)) + (3*b^2*d^2*x^2)/(a^3*d^3 - b^3*c^3 + 3*a*b^2* c^2*d - 3*a^2*b*c*d^2))/(x*(b*c^2 + 2*a*c*d) + a*c^2 + x^2*(a*d^2 + 2*b*c* d) + b*d^2*x^3) - (6*b^2*d*atanh((a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^ 3*b*c*d^3)/(a*d - b*c)^4 + (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3 *a^2*b*c*d^2))/(a*d - b*c)^4))/(a*d - b*c)^4