Integrand size = 15, antiderivative size = 143 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {b^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {3 b^2 d}{(b c-a d)^4 (a+b x)}+\frac {d^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {3 b d^2}{(b c-a d)^4 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 b^2 d^2 \log (c+d x)}{(b c-a d)^5} \]
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Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46} \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac {3 b^2 d}{(a+b x) (b c-a d)^4}-\frac {b^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {3 b d^2}{(c+d x) (b c-a d)^4}+\frac {d^2}{2 (c+d x)^2 (b c-a d)^3} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)^3}-\frac {3 b^3 d}{(b c-a d)^4 (a+b x)^2}+\frac {6 b^3 d^2}{(b c-a d)^5 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)^3}-\frac {3 b d^3}{(b c-a d)^4 (c+d x)^2}-\frac {6 b^2 d^3}{(b c-a d)^5 (c+d x)}\right ) \, dx \\ & = -\frac {b^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {3 b^2 d}{(b c-a d)^4 (a+b x)}+\frac {d^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {3 b d^2}{(b c-a d)^4 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 b^2 d^2 \log (c+d x)}{(b c-a d)^5} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {-\frac {b^2 (b c-a d)^2}{(a+b x)^2}+\frac {6 b^2 d (b c-a d)}{a+b x}+\frac {d^2 (b c-a d)^2}{(c+d x)^2}+\frac {6 b d^2 (b c-a d)}{c+d x}+12 b^2 d^2 \log (a+b x)-12 b^2 d^2 \log (c+d x)}{2 (b c-a d)^5} \]
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Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {d^{2}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {6 d^{2} b^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {3 d^{2} b}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {b^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {6 d^{2} b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {3 b^{2} d}{\left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(140\) |
risch | \(\frac {\frac {6 b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {9 b^{2} d^{2} \left (a d +b c \right ) x^{2}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {2 \left (a^{2} d^{2}+7 a b c d +b^{2} c^{2}\right ) b d x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {a^{3} d^{3}-7 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {6 b^{2} d^{2} \ln \left (-d x -c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {6 b^{2} d^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(477\) |
norman | \(\frac {\frac {\left (9 a \,b^{4} d^{5}+9 b^{5} c \,d^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {-a^{3} b^{2} d^{5}+7 a^{2} b^{3} c \,d^{4}+7 a \,b^{4} c^{2} d^{3}-b^{5} c^{3} d^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {6 b^{3} d^{3} x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {2 \left (a^{2} b^{3} d^{5}+7 a \,b^{4} c \,d^{4}+b^{5} c^{2} d^{3}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {6 b^{2} d^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {6 b^{2} d^{2} \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(517\) |
parallelrisch | \(-\frac {-8 a^{3} b^{3} c \,d^{5}+8 a \,b^{5} c^{3} d^{3}-12 x^{3} a \,b^{5} d^{6}+12 x^{3} b^{6} c \,d^{5}-18 x^{2} a^{2} b^{4} d^{6}+18 x^{2} b^{6} c^{2} d^{4}-4 x \,a^{3} b^{3} d^{6}+4 x \,b^{6} c^{3} d^{3}+12 \ln \left (b x +a \right ) x^{4} b^{6} d^{6}-12 \ln \left (d x +c \right ) x^{4} b^{6} d^{6}-24 x \,a^{2} b^{4} c \,d^{5}+24 x a \,b^{5} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a \,b^{5} d^{6}+24 \ln \left (b x +a \right ) x^{3} b^{6} c \,d^{5}-24 \ln \left (d x +c \right ) x^{3} a \,b^{5} d^{6}-24 \ln \left (d x +c \right ) x^{3} b^{6} c \,d^{5}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} d^{6}+12 \ln \left (b x +a \right ) x^{2} b^{6} c^{2} d^{4}-12 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} d^{6}-12 \ln \left (d x +c \right ) x^{2} b^{6} c^{2} d^{4}+12 \ln \left (b x +a \right ) a^{2} b^{4} c^{2} d^{4}-12 \ln \left (d x +c \right ) a^{2} b^{4} c^{2} d^{4}+48 \ln \left (b x +a \right ) x^{2} a \,b^{5} c \,d^{5}-48 \ln \left (d x +c \right ) x^{2} a \,b^{5} c \,d^{5}+24 \ln \left (b x +a \right ) x \,a^{2} b^{4} c \,d^{5}+24 \ln \left (b x +a \right ) x a \,b^{5} c^{2} d^{4}-24 \ln \left (d x +c \right ) x \,a^{2} b^{4} c \,d^{5}-24 \ln \left (d x +c \right ) x a \,b^{5} c^{2} d^{4}+a^{4} b^{2} d^{6}-b^{6} c^{4} d^{2}}{2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \left (d x +c \right )^{2} \left (b x +a \right )^{2} b^{2} d^{2}}\) | \(577\) |
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Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (139) = 278\).
Time = 0.24 (sec) , antiderivative size = 760, normalized size of antiderivative = 5.31 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {b^{4} c^{4} - 8 \, a b^{3} c^{3} d + 8 \, a^{3} b c d^{3} - a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 18 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d + 6 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x - 12 \, {\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (128) = 256\).
Time = 1.44 (sec) , antiderivative size = 881, normalized size of antiderivative = 6.16 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 b^{2} d^{2} \log {\left (x + \frac {- \frac {6 a^{6} b^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac {36 a^{5} b^{3} c d^{7}}{\left (a d - b c\right )^{5}} - \frac {90 a^{4} b^{4} c^{2} d^{6}}{\left (a d - b c\right )^{5}} + \frac {120 a^{3} b^{5} c^{3} d^{5}}{\left (a d - b c\right )^{5}} - \frac {90 a^{2} b^{6} c^{4} d^{4}}{\left (a d - b c\right )^{5}} + \frac {36 a b^{7} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + 6 a b^{2} d^{3} - \frac {6 b^{8} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + 6 b^{3} c d^{2}}{12 b^{3} d^{3}} \right )}}{\left (a d - b c\right )^{5}} - \frac {6 b^{2} d^{2} \log {\left (x + \frac {\frac {6 a^{6} b^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac {36 a^{5} b^{3} c d^{7}}{\left (a d - b c\right )^{5}} + \frac {90 a^{4} b^{4} c^{2} d^{6}}{\left (a d - b c\right )^{5}} - \frac {120 a^{3} b^{5} c^{3} d^{5}}{\left (a d - b c\right )^{5}} + \frac {90 a^{2} b^{6} c^{4} d^{4}}{\left (a d - b c\right )^{5}} - \frac {36 a b^{7} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + 6 a b^{2} d^{3} + \frac {6 b^{8} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + 6 b^{3} c d^{2}}{12 b^{3} d^{3}} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{3} d^{3} + 7 a^{2} b c d^{2} + 7 a b^{2} c^{2} d - b^{3} c^{3} + 12 b^{3} d^{3} x^{3} + x^{2} \cdot \left (18 a b^{2} d^{3} + 18 b^{3} c d^{2}\right ) + x \left (4 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{2 a^{6} c^{2} d^{4} - 8 a^{5} b c^{3} d^{3} + 12 a^{4} b^{2} c^{4} d^{2} - 8 a^{3} b^{3} c^{5} d + 2 a^{2} b^{4} c^{6} + x^{4} \cdot \left (2 a^{4} b^{2} d^{6} - 8 a^{3} b^{3} c d^{5} + 12 a^{2} b^{4} c^{2} d^{4} - 8 a b^{5} c^{3} d^{3} + 2 b^{6} c^{4} d^{2}\right ) + x^{3} \cdot \left (4 a^{5} b d^{6} - 12 a^{4} b^{2} c d^{5} + 8 a^{3} b^{3} c^{2} d^{4} + 8 a^{2} b^{4} c^{3} d^{3} - 12 a b^{5} c^{4} d^{2} + 4 b^{6} c^{5} d\right ) + x^{2} \cdot \left (2 a^{6} d^{6} - 18 a^{4} b^{2} c^{2} d^{4} + 32 a^{3} b^{3} c^{3} d^{3} - 18 a^{2} b^{4} c^{4} d^{2} + 2 b^{6} c^{6}\right ) + x \left (4 a^{6} c d^{5} - 12 a^{5} b c^{2} d^{4} + 8 a^{4} b^{2} c^{3} d^{3} + 8 a^{3} b^{3} c^{4} d^{2} - 12 a^{2} b^{4} c^{5} d + 4 a b^{5} c^{6}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (139) = 278\).
Time = 0.21 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, b^{2} d^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {6 \, b^{2} d^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {12 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (139) = 278\).
Time = 0.31 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.41 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, b^{3} d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {6 \, b^{2} d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {12 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 18 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
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Time = 0.59 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.79 \[ \int \frac {1}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {6\,b^3\,d^3\,x^3}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}-\frac {a^3\,d^3-7\,a^2\,b\,c\,d^2-7\,a\,b^2\,c^2\,d+b^3\,c^3}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {9\,b\,d\,x^2\,\left (c\,b^2\,d+a\,b\,d^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {2\,b\,d\,x\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {12\,b^2\,d^2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \]
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