Integrand size = 18, antiderivative size = 173 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {x}{b^2 d^3}+\frac {a^5}{b^3 (b c-a d)^3 (a+b x)}+\frac {c^5}{2 d^4 (b c-a d)^2 (c+d x)^2}-\frac {c^4 (3 b c-5 a d)}{d^4 (b c-a d)^3 (c+d x)}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^4} \]
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Time = 0.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {a^5}{b^3 (a+b x) (b c-a d)^3}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac {c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}-\frac {c^4 (3 b c-5 a d)}{d^4 (c+d x) (b c-a d)^3}+\frac {x}{b^2 d^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^2 d^3}-\frac {a^5}{b^2 (b c-a d)^3 (a+b x)^2}-\frac {a^4 (-5 b c+2 a d)}{b^2 (b c-a d)^4 (a+b x)}-\frac {c^5}{d^3 (-b c+a d)^2 (c+d x)^3}-\frac {c^4 (3 b c-5 a d)}{d^3 (-b c+a d)^3 (c+d x)^2}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right )}{d^3 (-b c+a d)^4 (c+d x)}\right ) \, dx \\ & = \frac {x}{b^2 d^3}+\frac {a^5}{b^3 (b c-a d)^3 (a+b x)}+\frac {c^5}{2 d^4 (b c-a d)^2 (c+d x)^2}-\frac {c^4 (3 b c-5 a d)}{d^4 (b c-a d)^3 (c+d x)}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^4} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {x}{b^2 d^3}+\frac {a^5}{b^3 (b c-a d)^3 (a+b x)}+\frac {c^5}{2 d^4 (b c-a d)^2 (c+d x)^2}+\frac {c^4 (3 b c-5 a d)}{d^4 (-b c+a d)^3 (c+d x)}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^4} \]
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Time = 0.52 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {x}{b^{2} d^{3}}-\frac {c^{4} \left (5 a d -3 b c \right )}{d^{4} \left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {c^{5}}{2 d^{4} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}-\frac {c^{3} \left (10 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{4}}-\frac {a^{5}}{b^{3} \left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {a^{4} \left (2 a d -5 b c \right ) \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{4}}\) | \(174\) |
norman | \(\frac {\frac {x^{4}}{b d}-\frac {\left (2 a^{5} d^{5}-a^{4} b c \,d^{4}-4 a^{2} b^{3} c^{3} d^{2}+12 a \,b^{4} c^{4} d -6 b^{5} c^{5}\right ) x^{2}}{d^{3} b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c \left (8 a^{5} d^{5}-4 a^{4} b c \,d^{4}-8 a^{3} b^{2} c^{2} d^{3}+18 a^{2} b^{3} c^{3} d^{2}+7 a \,b^{4} c^{4} d -9 b^{5} c^{5}\right ) x}{2 d^{4} b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{2} a \left (4 a^{4} d^{4}-2 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}+19 a \,b^{3} c^{3} d -9 b^{4} c^{4}\right )}{2 d^{4} b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {a^{4} \left (2 a d -5 b c \right ) \ln \left (b x +a \right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{3}}-\frac {c^{3} \left (10 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(504\) |
risch | \(\frac {x}{b^{2} d^{3}}+\frac {-\frac {\left (a^{5} d^{5}+5 a \,b^{4} c^{4} d -3 b^{5} c^{5}\right ) x^{2}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}-\frac {\left (4 a^{5} d^{5}+10 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -5 b^{5} c^{5}\right ) c x}{2 d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}-\frac {a \,c^{2} \left (2 a^{4} d^{4}+9 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right )}{2 b d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{b^{2} d^{3} \left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {10 c^{3} \ln \left (-d x -c \right ) a^{2}}{d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {10 c^{4} \ln \left (-d x -c \right ) a b}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {3 c^{5} \ln \left (-d x -c \right ) b^{2}}{d^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {2 a^{5} \ln \left (b x +a \right ) d}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{3}}+\frac {5 a^{4} \ln \left (b x +a \right ) c}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}\) | \(614\) |
parallelrisch | \(-\frac {40 \ln \left (d x +c \right ) x \,a^{3} b^{3} c^{4} d^{3}-20 \ln \left (d x +c \right ) x \,a^{2} b^{4} c^{5} d^{2}-8 \ln \left (d x +c \right ) x a \,b^{5} c^{6} d +9 x \,b^{6} c^{7}+4 a^{6} c^{2} d^{5}+9 a \,b^{5} c^{7}+4 x^{2} a^{6} d^{7}-6 a^{5} b \,c^{3} d^{4}-4 a^{4} b^{2} c^{4} d^{3}+25 a^{3} b^{3} c^{5} d^{2}-28 a^{2} b^{4} c^{6} d -2 x^{4} a^{4} b^{2} d^{7}-2 x^{4} b^{6} c^{4} d^{3}+12 x^{2} b^{6} c^{6} d +8 x \,a^{6} c \,d^{6}+4 \ln \left (b x +a \right ) x^{2} a^{6} d^{7}+6 \ln \left (d x +c \right ) x \,b^{6} c^{7}+4 \ln \left (b x +a \right ) a^{6} c^{2} d^{5}+6 \ln \left (d x +c \right ) a \,b^{5} c^{7}-6 x^{2} a^{5} b c \,d^{6}+2 x^{2} a^{4} b^{2} c^{2} d^{5}-8 x^{2} a^{3} b^{3} c^{3} d^{4}+32 x^{2} a^{2} b^{4} c^{4} d^{3}-36 x^{2} a \,b^{5} c^{5} d^{2}-12 x \,a^{5} b \,c^{2} d^{5}-4 x \,a^{4} b^{2} c^{3} d^{4}+26 x \,a^{3} b^{3} c^{4} d^{3}-11 x \,a^{2} b^{4} c^{5} d^{2}-16 x a \,b^{5} c^{6} d +4 \ln \left (b x +a \right ) x^{3} a^{5} b \,d^{7}+6 \ln \left (d x +c \right ) x^{3} b^{6} c^{5} d^{2}+12 \ln \left (d x +c \right ) x^{2} b^{6} c^{6} d +8 \ln \left (b x +a \right ) x \,a^{6} c \,d^{6}-10 \ln \left (b x +a \right ) a^{5} b \,c^{3} d^{4}+20 \ln \left (d x +c \right ) a^{3} b^{3} c^{5} d^{2}-20 \ln \left (d x +c \right ) a^{2} b^{4} c^{6} d +8 x^{4} a^{3} b^{3} c \,d^{6}-12 x^{4} a^{2} b^{4} c^{2} d^{5}+8 x^{4} a \,b^{5} c^{3} d^{4}-10 \ln \left (b x +a \right ) x^{3} a^{4} b^{2} c \,d^{6}+20 \ln \left (d x +c \right ) x^{3} a^{2} b^{4} c^{3} d^{4}-20 \ln \left (d x +c \right ) x^{3} a \,b^{5} c^{4} d^{3}-2 \ln \left (b x +a \right ) x^{2} a^{5} b c \,d^{6}-20 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} c^{2} d^{5}+20 \ln \left (d x +c \right ) x^{2} a^{3} b^{3} c^{3} d^{4}+20 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} c^{4} d^{3}-34 \ln \left (d x +c \right ) x^{2} a \,b^{5} c^{5} d^{2}-16 \ln \left (b x +a \right ) x \,a^{5} b \,c^{2} d^{5}-10 \ln \left (b x +a \right ) x \,a^{4} b^{2} c^{3} d^{4}}{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^{3} d^{4}}\) | \(877\) |
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Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (171) = 342\).
Time = 0.27 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.67 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {5 \, a b^{5} c^{7} - 14 \, a^{2} b^{4} c^{6} d + 9 \, a^{3} b^{3} c^{5} d^{2} - 2 \, a^{5} b c^{3} d^{4} + 2 \, a^{6} c^{2} d^{5} - 2 \, {\left (b^{6} c^{4} d^{3} - 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} - 4 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{4} - 2 \, {\left (2 \, b^{6} c^{5} d^{2} - 7 \, a b^{5} c^{4} d^{3} + 8 \, a^{2} b^{4} c^{3} d^{4} - 2 \, a^{3} b^{3} c^{2} d^{5} - 2 \, a^{4} b^{2} c d^{6} + a^{5} b d^{7}\right )} x^{3} + 2 \, {\left (2 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 7 \, a^{2} b^{4} c^{4} d^{3} - 8 \, a^{3} b^{3} c^{3} d^{4} + 7 \, a^{4} b^{2} c^{2} d^{5} - 3 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x^{2} + {\left (5 \, b^{6} c^{7} - 10 \, a b^{5} c^{6} d + a^{2} b^{4} c^{5} d^{2} - 2 \, a^{3} b^{3} c^{4} d^{3} + 8 \, a^{4} b^{2} c^{3} d^{4} - 6 \, a^{5} b c^{2} d^{5} + 4 \, a^{6} c d^{6}\right )} x - 2 \, {\left (5 \, a^{5} b c^{3} d^{4} - 2 \, a^{6} c^{2} d^{5} + {\left (5 \, a^{4} b^{2} c d^{6} - 2 \, a^{5} b d^{7}\right )} x^{3} + {\left (10 \, a^{4} b^{2} c^{2} d^{5} + a^{5} b c d^{6} - 2 \, a^{6} d^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{2} c^{3} d^{4} + 8 \, a^{5} b c^{2} d^{5} - 4 \, a^{6} c d^{6}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, a b^{5} c^{7} - 10 \, a^{2} b^{4} c^{6} d + 10 \, a^{3} b^{3} c^{5} d^{2} + {\left (3 \, b^{6} c^{5} d^{2} - 10 \, a b^{5} c^{4} d^{3} + 10 \, a^{2} b^{4} c^{3} d^{4}\right )} x^{3} + {\left (6 \, b^{6} c^{6} d - 17 \, a b^{5} c^{5} d^{2} + 10 \, a^{2} b^{4} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{3} d^{4}\right )} x^{2} + {\left (3 \, b^{6} c^{7} - 4 \, a b^{5} c^{6} d - 10 \, a^{2} b^{4} c^{5} d^{2} + 20 \, a^{3} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{7} c^{6} d^{4} - 4 \, a^{2} b^{6} c^{5} d^{5} + 6 \, a^{3} b^{5} c^{4} d^{6} - 4 \, a^{4} b^{4} c^{3} d^{7} + a^{5} b^{3} c^{2} d^{8} + {\left (b^{8} c^{4} d^{6} - 4 \, a b^{7} c^{3} d^{7} + 6 \, a^{2} b^{6} c^{2} d^{8} - 4 \, a^{3} b^{5} c d^{9} + a^{4} b^{4} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{5} d^{5} - 7 \, a b^{7} c^{4} d^{6} + 8 \, a^{2} b^{6} c^{3} d^{7} - 2 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9} + a^{5} b^{3} d^{10}\right )} x^{2} + {\left (b^{8} c^{6} d^{4} - 2 \, a b^{7} c^{5} d^{5} - 2 \, a^{2} b^{6} c^{4} d^{6} + 8 \, a^{3} b^{5} c^{3} d^{7} - 7 \, a^{4} b^{4} c^{2} d^{8} + 2 \, a^{5} b^{3} c d^{9}\right )} x\right )}} \]
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Timed out. \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (171) = 342\).
Time = 0.21 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.05 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {{\left (5 \, a^{4} b c - 2 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}} - \frac {{\left (3 \, b^{2} c^{5} - 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}} - \frac {5 \, a b^{4} c^{6} - 9 \, a^{2} b^{3} c^{5} d - 2 \, a^{5} c^{2} d^{4} + 2 \, {\left (3 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - a^{5} d^{6}\right )} x^{2} + {\left (5 \, b^{5} c^{6} - 3 \, a b^{4} c^{5} d - 10 \, a^{2} b^{3} c^{4} d^{2} - 4 \, a^{5} c d^{5}\right )} x}{2 \, {\left (a b^{6} c^{5} d^{4} - 3 \, a^{2} b^{5} c^{4} d^{5} + 3 \, a^{3} b^{4} c^{3} d^{6} - a^{4} b^{3} c^{2} d^{7} + {\left (b^{7} c^{3} d^{6} - 3 \, a b^{6} c^{2} d^{7} + 3 \, a^{2} b^{5} c d^{8} - a^{3} b^{4} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{4} d^{5} - 5 \, a b^{6} c^{3} d^{6} + 3 \, a^{2} b^{5} c^{2} d^{7} + a^{3} b^{4} c d^{8} - a^{4} b^{3} d^{9}\right )} x^{2} + {\left (b^{7} c^{5} d^{4} - a b^{6} c^{4} d^{5} - 3 \, a^{2} b^{5} c^{3} d^{6} + 5 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8}\right )} x\right )}} + \frac {x}{b^{2} d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.87 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {a^{5} b^{4}}{{\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} {\left (b x + a\right )}} - \frac {{\left (3 \, b^{3} c^{5} - 10 \, a b^{2} c^{4} d + 10 \, a^{2} b c^{3} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{4} - 4 \, a b^{4} c^{3} d^{5} + 6 \, a^{2} b^{3} c^{2} d^{6} - 4 \, a^{3} b^{2} c d^{7} + a^{4} b d^{8}} + \frac {{\left (3 \, b c + 2 \, a d\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3} d^{4}} + \frac {{\left (2 \, b^{4} c^{4} d^{3} - 8 \, a b^{3} c^{3} d^{4} + 12 \, a^{2} b^{2} c^{2} d^{5} - 8 \, a^{3} b c d^{6} + 2 \, a^{4} d^{7} + \frac {9 \, b^{6} c^{5} d^{2} - 30 \, a b^{5} c^{4} d^{3} + 40 \, a^{2} b^{4} c^{3} d^{4} - 40 \, a^{3} b^{3} c^{2} d^{5} + 20 \, a^{4} b^{2} c d^{6} - 4 \, a^{5} b d^{7}}{{\left (b x + a\right )} b} + \frac {2 \, {\left (3 \, b^{8} c^{6} d - 13 \, a b^{7} c^{5} d^{2} + 20 \, a^{2} b^{6} c^{4} d^{3} - 20 \, a^{3} b^{5} c^{3} d^{4} + 15 \, a^{4} b^{4} c^{2} d^{5} - 6 \, a^{5} b^{3} c d^{6} + a^{6} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}}{2 \, {\left (b c - a d\right )}^{4} b^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{4}} \]
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Time = 0.77 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.82 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx=\frac {x}{b^2\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-10\,a\,b\,c^4\,d+3\,b^2\,c^5\right )}{a^4\,d^8-4\,a^3\,b\,c\,d^7+6\,a^2\,b^2\,c^2\,d^6-4\,a\,b^3\,c^3\,d^5+b^4\,c^4\,d^4}-\frac {\frac {x^2\,\left (a^5\,d^5+5\,a\,b^4\,c^4\,d-3\,b^5\,c^5\right )}{b\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^4\,d^4+9\,a\,b^3\,c^3\,d-5\,b^4\,c^4\right )}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^5\,d^5+10\,a^2\,b^3\,c^3\,d^2+3\,a\,b^4\,c^4\,d-5\,b^5\,c^5\right )}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^2\,\left (2\,c\,b^3\,d^4+a\,b^2\,d^5\right )+x\,\left (b^3\,c^2\,d^3+2\,a\,b^2\,c\,d^4\right )+b^3\,d^5\,x^3+a\,b^2\,c^2\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (2\,a^5\,d-5\,a^4\,b\,c\right )}{a^4\,b^3\,d^4-4\,a^3\,b^4\,c\,d^3+6\,a^2\,b^5\,c^2\,d^2-4\,a\,b^6\,c^3\,d+b^7\,c^4} \]
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