Integrand size = 15, antiderivative size = 186 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^7 x}{b^7}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) \log (a+b x)}{b^8} \]
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Time = 0.12 (sec) , antiderivative size = 186, 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 = {45} \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac {d^7 x}{b^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^7}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^7}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^6}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^5}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^4}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^3}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^2}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)}\right ) \, dx \\ & = \frac {d^7 x}{b^7}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) \log (a+b x)}{b^8} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(390\) vs. \(2(186)=372\).
Time = 0.12 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=-\frac {669 a^7 d^7+3 a^6 b d^6 (-343 c+1198 d x)+3 a^5 b^2 d^5 \left (70 c^2-1918 c d x+2575 d^2 x^2\right )+5 a^4 b^3 d^4 \left (14 c^3+252 c^2 d x-2625 c d^2 x^2+1640 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+84 c^3 d x+630 c^2 d^2 x^2-3080 c d^3 x^3+810 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+70 c^4 d x+350 c^3 d^2 x^2+1400 c^2 d^3 x^3-3150 c d^4 x^4+120 d^5 x^5\right )+a b^6 d \left (14 c^6+126 c^5 d x+525 c^4 d^2 x^2+1400 c^3 d^3 x^3+3150 c^2 d^4 x^4-2520 c d^5 x^5-360 d^6 x^6\right )+b^7 \left (10 c^7+84 c^6 d x+315 c^5 d^2 x^2+700 c^4 d^3 x^3+1050 c^3 d^4 x^4+1260 c^2 d^5 x^5-60 d^7 x^7\right )+420 d^6 (-b c+a d) (a+b x)^6 \log (a+b x)}{60 b^8 (a+b x)^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(451\) vs. \(2(176)=352\).
Time = 0.22 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.43
method | result | size |
risch | \(\frac {d^{7} x}{b^{7}}+\frac {\left (-21 a^{2} b^{4} d^{7}+42 a \,b^{5} c \,d^{6}-21 b^{6} c^{2} d^{5}\right ) x^{5}-\frac {35 b^{3} d^{4} \left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{2}-\frac {35 b^{2} d^{3} \left (13 a^{4} d^{4}-22 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{3}}{3}-\frac {7 b \,d^{2} \left (77 a^{5} d^{5}-125 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{2}}{4}-\frac {7 d \left (87 a^{6} d^{6}-137 a^{5} b c \,d^{5}+30 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +2 b^{6} c^{6}\right ) x}{10}-\frac {669 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b}}{b^{7} \left (b x +a \right )^{6}}-\frac {7 d^{7} \ln \left (b x +a \right ) a}{b^{8}}+\frac {7 d^{6} \ln \left (b x +a \right ) c}{b^{7}}\) | \(452\) |
default | \(\frac {d^{7} x}{b^{7}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 b^{8} \left (b x +a \right )^{3}}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{6 b^{8} \left (b x +a \right )^{6}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{4 b^{8} \left (b x +a \right )^{4}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{5 b^{8} \left (b x +a \right )^{5}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )}\) | \(456\) |
norman | \(\frac {\frac {d^{7} x^{7}}{b}-\frac {1029 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b^{8}}-\frac {3 \left (14 a^{2} d^{7}-14 a b c \,d^{6}+7 b^{2} c^{2} d^{5}\right ) x^{5}}{b^{3}}-\frac {5 \left (63 a^{3} d^{7}-63 a^{2} b c \,d^{6}+21 a \,b^{2} c^{2} d^{5}+7 b^{3} c^{3} d^{4}\right ) x^{4}}{2 b^{4}}-\frac {5 \left (154 a^{4} d^{7}-154 a^{3} b c \,d^{6}+42 a^{2} b^{2} c^{2} d^{5}+14 a \,b^{3} c^{3} d^{4}+7 b^{4} c^{4} d^{3}\right ) x^{3}}{3 b^{5}}-\frac {\left (875 a^{5} d^{7}-875 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+70 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{2}}{4 b^{6}}-\frac {\left (959 a^{6} d^{7}-959 a^{5} b c \,d^{6}+210 a^{4} b^{2} c^{2} d^{5}+70 a^{3} b^{3} c^{3} d^{4}+35 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+14 b^{6} c^{6} d \right ) x}{10 b^{7}}}{\left (b x +a \right )^{6}}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}\) | \(457\) |
parallelrisch | \(-\frac {2520 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}+14 a \,b^{6} c^{6} d +21 a^{2} b^{5} c^{5} d^{2}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}-2520 \ln \left (b x +a \right ) x^{5} a \,b^{6} c \,d^{6}-8400 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}-6300 \ln \left (b x +a \right ) x^{2} a^{4} b^{3} c \,d^{6}-6300 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c \,d^{6}-2520 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{6}+10 b^{7} c^{7}+1029 a^{7} d^{7}+6300 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}+1050 x^{2} a^{2} b^{5} c^{3} d^{4}+525 x^{2} a \,b^{6} c^{4} d^{3}-15400 x^{3} a^{3} b^{4} c \,d^{6}+4200 x^{3} a^{2} b^{5} c^{2} d^{5}+1400 x^{3} a \,b^{6} c^{3} d^{4}-9450 x^{4} a^{2} b^{5} c \,d^{6}-420 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+3150 x^{4} a \,b^{6} c^{2} d^{5}-2520 x^{5} a \,b^{6} c \,d^{6}-5754 x \,a^{5} b^{2} c \,d^{6}+1260 x \,a^{4} b^{3} c^{2} d^{5}+420 x \,a^{3} b^{4} c^{3} d^{4}+210 x \,a^{2} b^{5} c^{4} d^{3}+126 x a \,b^{6} c^{5} d^{2}-13125 x^{2} a^{4} b^{3} c \,d^{6}+3150 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (b x +a \right ) a^{7} d^{7}-60 x^{7} d^{7} b^{7}+6300 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}+2520 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} d^{7}+420 \ln \left (b x +a \right ) x^{6} a \,b^{6} d^{7}-420 \ln \left (b x +a \right ) x^{6} b^{7} c \,d^{6}+2520 x^{5} a^{2} b^{5} d^{7}+1260 x^{5} b^{7} c^{2} d^{5}+5754 x \,a^{6} b \,d^{7}+84 x \,b^{7} c^{6} d +13125 x^{2} a^{5} b^{2} d^{7}+315 x^{2} b^{7} c^{5} d^{2}+15400 x^{3} a^{4} b^{3} d^{7}+700 x^{3} b^{7} c^{4} d^{3}+9450 x^{4} a^{3} b^{4} d^{7}+1050 x^{4} b^{7} c^{3} d^{4}+8400 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}}{60 b^{8} \left (b x +a \right )^{6}}\) | \(737\) |
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Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (176) = 352\).
Time = 0.23 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {60 \, b^{7} d^{7} x^{7} + 360 \, a b^{6} d^{7} x^{6} - 10 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 70 \, a^{4} b^{3} c^{3} d^{4} - 210 \, a^{5} b^{2} c^{2} d^{5} + 1029 \, a^{6} b c d^{6} - 669 \, a^{7} d^{7} - 180 \, {\left (7 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + 2 \, a^{2} b^{5} d^{7}\right )} x^{5} - 150 \, {\left (7 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} - 63 \, a^{2} b^{5} c d^{6} + 27 \, a^{3} b^{4} d^{7}\right )} x^{4} - 100 \, {\left (7 \, b^{7} c^{4} d^{3} + 14 \, a b^{6} c^{3} d^{4} + 42 \, a^{2} b^{5} c^{2} d^{5} - 154 \, a^{3} b^{4} c d^{6} + 82 \, a^{4} b^{3} d^{7}\right )} x^{3} - 15 \, {\left (21 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 70 \, a^{2} b^{5} c^{3} d^{4} + 210 \, a^{3} b^{4} c^{2} d^{5} - 875 \, a^{4} b^{3} c d^{6} + 515 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \, {\left (14 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 70 \, a^{3} b^{4} c^{3} d^{4} + 210 \, a^{4} b^{3} c^{2} d^{5} - 959 \, a^{5} b^{2} c d^{6} + 599 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 6 \, {\left (a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 15 \, {\left (a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 20 \, {\left (a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 15 \, {\left (a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 6 \, {\left (a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (176) = 352\).
Time = 0.23 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.77 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^{7} x}{b^{7}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (176) = 352\).
Time = 0.29 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.47 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^{7} x}{b^{7}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b x + a\right )}^{6} b^{8}} \]
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Time = 0.40 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^7\,x}{b^7}-\frac {\ln \left (a+b\,x\right )\,\left (7\,a\,d^7-7\,b\,c\,d^6\right )}{b^8}-\frac {\frac {669\,a^7\,d^7-1029\,a^6\,b\,c\,d^6+210\,a^5\,b^2\,c^2\,d^5+70\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+14\,a\,b^6\,c^6\,d+10\,b^7\,c^7}{60\,b}+x\,\left (\frac {609\,a^6\,d^7}{10}-\frac {959\,a^5\,b\,c\,d^6}{10}+21\,a^4\,b^2\,c^2\,d^5+7\,a^3\,b^3\,c^3\,d^4+\frac {7\,a^2\,b^4\,c^4\,d^3}{2}+\frac {21\,a\,b^5\,c^5\,d^2}{10}+\frac {7\,b^6\,c^6\,d}{5}\right )+x^3\,\left (\frac {455\,a^4\,b^2\,d^7}{3}-\frac {770\,a^3\,b^3\,c\,d^6}{3}+70\,a^2\,b^4\,c^2\,d^5+\frac {70\,a\,b^5\,c^3\,d^4}{3}+\frac {35\,b^6\,c^4\,d^3}{3}\right )+x^2\,\left (\frac {539\,a^5\,b\,d^7}{4}-\frac {875\,a^4\,b^2\,c\,d^6}{4}+\frac {105\,a^3\,b^3\,c^2\,d^5}{2}+\frac {35\,a^2\,b^4\,c^3\,d^4}{2}+\frac {35\,a\,b^5\,c^4\,d^3}{4}+\frac {21\,b^6\,c^5\,d^2}{4}\right )+x^5\,\left (21\,a^2\,b^4\,d^7-42\,a\,b^5\,c\,d^6+21\,b^6\,c^2\,d^5\right )+x^4\,\left (\frac {175\,a^3\,b^3\,d^7}{2}-\frac {315\,a^2\,b^4\,c\,d^6}{2}+\frac {105\,a\,b^5\,c^2\,d^5}{2}+\frac {35\,b^6\,c^3\,d^4}{2}\right )}{a^6\,b^7+6\,a^5\,b^8\,x+15\,a^4\,b^9\,x^2+20\,a^3\,b^{10}\,x^3+15\,a^2\,b^{11}\,x^4+6\,a\,b^{12}\,x^5+b^{13}\,x^6} \]
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