Integrand size = 15, antiderivative size = 194 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {(b c-a d)^7}{7 d^8 (c+d x)^7}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {7 b^6 (b c-a d)}{d^8 (c+d x)}+\frac {b^7 \log (c+d x)}{d^8} \]
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Time = 0.16 (sec) , antiderivative size = 194, 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} \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {7 b^6 (b c-a d)}{d^8 (c+d x)}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {(b c-a d)^7}{7 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^7}{d^7 (c+d x)^8}+\frac {7 b (b c-a d)^6}{d^7 (c+d x)^7}-\frac {21 b^2 (b c-a d)^5}{d^7 (c+d x)^6}+\frac {35 b^3 (b c-a d)^4}{d^7 (c+d x)^5}-\frac {35 b^4 (b c-a d)^3}{d^7 (c+d x)^4}+\frac {21 b^5 (b c-a d)^2}{d^7 (c+d x)^3}-\frac {7 b^6 (b c-a d)}{d^7 (c+d x)^2}+\frac {b^7}{d^7 (c+d x)}\right ) \, dx \\ & = \frac {(b c-a d)^7}{7 d^8 (c+d x)^7}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {7 b^6 (b c-a d)}{d^8 (c+d x)}+\frac {b^7 \log (c+d x)}{d^8} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {(b c-a d) \left (60 a^6 d^6+10 a^5 b d^5 (13 c+49 d x)+2 a^4 b^2 d^4 \left (107 c^2+539 c d x+882 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+1813 c^2 d x+3969 c d^2 x^2+3675 d^3 x^3\right )+a^2 b^4 d^2 \left (459 c^4+2793 c^3 d x+6909 c^2 d^2 x^2+8575 c d^3 x^3+4900 d^4 x^4\right )+a b^5 d \left (669 c^5+4263 c^4 d x+11319 c^3 d^2 x^2+15925 c^2 d^3 x^3+12250 c d^4 x^4+4410 d^5 x^5\right )+b^6 \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(182)=364\).
Time = 0.23 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.30
method | result | size |
risch | \(\frac {-\frac {7 b^{6} \left (a d -b c \right ) x^{6}}{d^{2}}-\frac {21 b^{5} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) x^{5}}{2 d^{3}}-\frac {35 b^{4} \left (2 a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -11 b^{3} c^{3}\right ) x^{4}}{6 d^{4}}-\frac {35 b^{3} \left (3 a^{4} d^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 b^{4} c^{4}\right ) x^{3}}{12 d^{5}}-\frac {7 b^{2} \left (12 a^{5} d^{5}+15 a^{4} b c \,d^{4}+20 a^{3} b^{2} c^{2} d^{3}+30 a^{2} b^{3} c^{3} d^{2}+60 a \,b^{4} c^{4} d -137 b^{5} c^{5}\right ) x^{2}}{20 d^{6}}-\frac {7 b \left (10 a^{6} d^{6}+12 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+30 a^{2} b^{4} c^{4} d^{2}+60 a \,b^{5} c^{5} d -147 b^{6} c^{6}\right ) x}{60 d^{7}}-\frac {60 a^{7} d^{7}+70 a^{6} b c \,d^{6}+84 a^{5} b^{2} c^{2} d^{5}+105 a^{4} b^{3} c^{3} d^{4}+140 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}+420 a \,b^{6} c^{6} d -1089 b^{7} c^{7}}{420 d^{8}}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}\) | \(446\) |
norman | \(\frac {-\frac {60 a^{7} d^{7}+70 a^{6} b c \,d^{6}+84 a^{5} b^{2} c^{2} d^{5}+105 a^{4} b^{3} c^{3} d^{4}+140 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}+420 a \,b^{6} c^{6} d -1089 b^{7} c^{7}}{420 d^{8}}-\frac {7 \left (a \,b^{6} d -b^{7} c \right ) x^{6}}{d^{2}}-\frac {21 \left (a^{2} b^{5} d^{2}+2 a \,b^{6} c d -3 b^{7} c^{2}\right ) x^{5}}{2 d^{3}}-\frac {35 \left (2 a^{3} b^{4} d^{3}+3 a^{2} b^{5} c \,d^{2}+6 a \,b^{6} c^{2} d -11 b^{7} c^{3}\right ) x^{4}}{6 d^{4}}-\frac {35 \left (3 a^{4} b^{3} d^{4}+4 a^{3} b^{4} c \,d^{3}+6 a^{2} b^{5} c^{2} d^{2}+12 a \,b^{6} c^{3} d -25 b^{7} c^{4}\right ) x^{3}}{12 d^{5}}-\frac {7 \left (12 a^{5} b^{2} d^{5}+15 a^{4} b^{3} c \,d^{4}+20 a^{3} b^{4} c^{2} d^{3}+30 a^{2} b^{5} c^{3} d^{2}+60 a \,b^{6} c^{4} d -137 b^{7} c^{5}\right ) x^{2}}{20 d^{6}}-\frac {7 \left (10 a^{6} b \,d^{6}+12 a^{5} b^{2} c \,d^{5}+15 a^{4} b^{3} c^{2} d^{4}+20 a^{3} b^{4} c^{3} d^{3}+30 a^{2} b^{5} c^{4} d^{2}+60 a \,b^{6} c^{5} d -147 b^{7} c^{6}\right ) x}{60 d^{7}}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}\) | \(458\) |
default | \(-\frac {21 b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{8} \left (d x +c \right )^{2}}-\frac {21 b^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{5 d^{8} \left (d x +c \right )^{5}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}-\frac {7 b \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{6 d^{8} \left (d x +c \right )^{6}}-\frac {a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}{7 d^{8} \left (d x +c \right )^{7}}-\frac {35 b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{8} \left (d x +c \right )^{3}}-\frac {7 b^{6} \left (a d -b c \right )}{d^{8} \left (d x +c \right )}-\frac {35 b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{4 d^{8} \left (d x +c \right )^{4}}\) | \(462\) |
parallelrisch | \(\frac {2940 \ln \left (d x +c \right ) x^{6} b^{7} c \,d^{6}+8820 \ln \left (d x +c \right ) x^{5} b^{7} c^{2} d^{5}+14700 \ln \left (d x +c \right ) x^{4} b^{7} c^{3} d^{4}+14700 \ln \left (d x +c \right ) x^{3} b^{7} c^{4} d^{3}+8820 \ln \left (d x +c \right ) x^{2} b^{7} c^{5} d^{2}+2940 \ln \left (d x +c \right ) x \,b^{7} c^{6} d -420 a \,b^{6} c^{6} d -210 a^{2} b^{5} c^{5} d^{2}-105 a^{4} b^{3} c^{3} d^{4}-140 a^{3} b^{4} c^{4} d^{3}-70 a^{6} b c \,d^{6}-84 a^{5} b^{2} c^{2} d^{5}+1089 b^{7} c^{7}-60 a^{7} d^{7}-4410 x^{2} a^{2} b^{5} c^{3} d^{4}-8820 x^{2} a \,b^{6} c^{4} d^{3}-4900 x^{3} a^{3} b^{4} c \,d^{6}-7350 x^{3} a^{2} b^{5} c^{2} d^{5}-14700 x^{3} a \,b^{6} c^{3} d^{4}-7350 x^{4} a^{2} b^{5} c \,d^{6}-14700 x^{4} a \,b^{6} c^{2} d^{5}-8820 x^{5} a \,b^{6} c \,d^{6}-588 x \,a^{5} b^{2} c \,d^{6}-735 x \,a^{4} b^{3} c^{2} d^{5}-980 x \,a^{3} b^{4} c^{3} d^{4}-1470 x \,a^{2} b^{5} c^{4} d^{3}-2940 x a \,b^{6} c^{5} d^{2}-2205 x^{2} a^{4} b^{3} c \,d^{6}-2940 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (d x +c \right ) x^{7} b^{7} d^{7}+420 \ln \left (d x +c \right ) b^{7} c^{7}-4410 x^{5} a^{2} b^{5} d^{7}+13230 x^{5} b^{7} c^{2} d^{5}-2940 x^{6} a \,b^{6} d^{7}-490 x \,a^{6} b \,d^{7}+7203 x \,b^{7} c^{6} d -1764 x^{2} a^{5} b^{2} d^{7}+20139 x^{2} b^{7} c^{5} d^{2}-3675 x^{3} a^{4} b^{3} d^{7}+30625 x^{3} b^{7} c^{4} d^{3}-4900 x^{4} a^{3} b^{4} d^{7}+26950 x^{4} b^{7} c^{3} d^{4}+2940 x^{6} b^{7} c \,d^{6}}{420 d^{8} \left (d x +c \right )^{7}}\) | \(632\) |
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Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (182) = 364\).
Time = 0.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.22 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x + 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, b^{7} c d^{6} x^{6} + 21 \, b^{7} c^{2} d^{5} x^{5} + 35 \, b^{7} c^{3} d^{4} x^{4} + 35 \, b^{7} c^{4} d^{3} x^{3} + 21 \, b^{7} c^{5} d^{2} x^{2} + 7 \, b^{7} c^{6} d x + b^{7} c^{7}\right )} \log \left (d x + c\right )}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (182) = 364\).
Time = 0.22 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} + \frac {b^{7} \log \left (d x + c\right )}{d^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (182) = 364\).
Time = 0.31 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {b^{7} \log \left ({\left | d x + c \right |}\right )}{d^{8}} + \frac {2940 \, {\left (b^{7} c d^{5} - a b^{6} d^{6}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{4} - 2 \, a b^{6} c d^{5} - a^{2} b^{5} d^{6}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{3} - 6 \, a b^{6} c^{2} d^{4} - 3 \, a^{2} b^{5} c d^{5} - 2 \, a^{3} b^{4} d^{6}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{2} - 12 \, a b^{6} c^{3} d^{3} - 6 \, a^{2} b^{5} c^{2} d^{4} - 4 \, a^{3} b^{4} c d^{5} - 3 \, a^{4} b^{3} d^{6}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d - 60 \, a b^{6} c^{4} d^{2} - 30 \, a^{2} b^{5} c^{3} d^{3} - 20 \, a^{3} b^{4} c^{2} d^{4} - 15 \, a^{4} b^{3} c d^{5} - 12 \, a^{5} b^{2} d^{6}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} - 60 \, a b^{6} c^{5} d - 30 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} - 15 \, a^{4} b^{3} c^{2} d^{4} - 12 \, a^{5} b^{2} c d^{5} - 10 \, a^{6} b d^{6}\right )} x + \frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7}}{d}}{420 \, {\left (d x + c\right )}^{7} d^{7}} \]
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Time = 0.50 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.37 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {b^7\,\ln \left (c+d\,x\right )}{d^8}-\frac {x\,\left (\frac {7\,a^6\,b\,d^7}{6}+\frac {7\,a^5\,b^2\,c\,d^6}{5}+\frac {7\,a^4\,b^3\,c^2\,d^5}{4}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{2}+7\,a\,b^6\,c^5\,d^2-\frac {343\,b^7\,c^6\,d}{20}\right )+x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (\frac {35\,a^4\,b^3\,d^7}{4}+\frac {35\,a^3\,b^4\,c\,d^6}{3}+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+35\,a\,b^6\,c^3\,d^4-\frac {875\,b^7\,c^4\,d^3}{12}\right )+x^5\,\left (\frac {21\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6-\frac {63\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (\frac {21\,a^5\,b^2\,d^7}{5}+\frac {21\,a^4\,b^3\,c\,d^6}{4}+7\,a^3\,b^4\,c^2\,d^5+\frac {21\,a^2\,b^5\,c^3\,d^4}{2}+21\,a\,b^6\,c^4\,d^3-\frac {959\,b^7\,c^5\,d^2}{20}\right )+\frac {a^7\,d^7}{7}-\frac {363\,b^7\,c^7}{140}+x^4\,\left (\frac {35\,a^3\,b^4\,d^7}{3}+\frac {35\,a^2\,b^5\,c\,d^6}{2}+35\,a\,b^6\,c^2\,d^5-\frac {385\,b^7\,c^3\,d^4}{6}\right )+\frac {a^2\,b^5\,c^5\,d^2}{2}+\frac {a^3\,b^4\,c^4\,d^3}{3}+\frac {a^4\,b^3\,c^3\,d^4}{4}+\frac {a^5\,b^2\,c^2\,d^5}{5}+a\,b^6\,c^6\,d+\frac {a^6\,b\,c\,d^6}{6}}{d^8\,{\left (c+d\,x\right )}^7} \]
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