Integrand size = 15, antiderivative size = 194 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac {d^7 \log (a+b x)}{b^8} \]
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Time = 0.10 (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} \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^8}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^7}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^6}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^5}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^4}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^3}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^2}+\frac {d^7}{b^7 (a+b x)}\right ) \, dx \\ & = -\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac {d^7 \log (a+b x)}{b^8} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.59 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {(b c-a d) \left (1089 a^6 d^6+3 a^5 b d^5 (223 c+2401 d x)+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+2793 c^2 d x+11319 c d^2 x^2+30625 d^3 x^3\right )+a^2 b^4 d^2 \left (214 c^4+1813 c^3 d x+6909 c^2 d^2 x^2+15925 c d^3 x^3+26950 d^4 x^4\right )+a b^5 d \left (130 c^5+1078 c^4 d x+3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (60 c^6+490 c^5 d x+1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(446\) vs. \(2(182)=364\).
Time = 0.22 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.30
method | result | size |
risch | \(\frac {\frac {7 d^{6} \left (a d -b c \right ) x^{6}}{b^{2}}+\frac {21 d^{5} \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{5}}{2 b^{3}}+\frac {35 d^{4} \left (11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{4}}{6 b^{4}}+\frac {35 d^{3} \left (25 a^{4} d^{4}-12 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{3}}{12 b^{5}}+\frac {7 d^{2} \left (137 a^{5} d^{5}-60 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-15 a \,b^{4} c^{4} d -12 b^{5} c^{5}\right ) x^{2}}{20 b^{6}}+\frac {7 d \left (147 a^{6} d^{6}-60 a^{5} b c \,d^{5}-30 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-15 a^{2} b^{4} c^{4} d^{2}-12 a \,b^{5} c^{5} d -10 b^{6} c^{6}\right ) x}{60 b^{7}}+\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\) | \(447\) |
norman | \(\frac {\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}+\frac {7 \left (a \,d^{7}-b c \,d^{6}\right ) x^{6}}{b^{2}}+\frac {21 \left (3 a^{2} d^{7}-2 a b c \,d^{6}-b^{2} c^{2} d^{5}\right ) x^{5}}{2 b^{3}}+\frac {35 \left (11 a^{3} d^{7}-6 a^{2} b c \,d^{6}-3 a \,b^{2} c^{2} d^{5}-2 b^{3} c^{3} d^{4}\right ) x^{4}}{6 b^{4}}+\frac {35 \left (25 a^{4} d^{7}-12 a^{3} b c \,d^{6}-6 a^{2} b^{2} c^{2} d^{5}-4 a \,b^{3} c^{3} d^{4}-3 b^{4} c^{4} d^{3}\right ) x^{3}}{12 b^{5}}+\frac {7 \left (137 a^{5} d^{7}-60 a^{4} b c \,d^{6}-30 a^{3} b^{2} c^{2} d^{5}-20 a^{2} b^{3} c^{3} d^{4}-15 a \,b^{4} c^{4} d^{3}-12 b^{5} c^{5} d^{2}\right ) x^{2}}{20 b^{6}}+\frac {7 \left (147 a^{6} d^{7}-60 a^{5} b c \,d^{6}-30 a^{4} b^{2} c^{2} d^{5}-20 a^{3} b^{3} c^{3} d^{4}-15 a^{2} b^{4} c^{4} d^{3}-12 a \,b^{5} c^{5} d^{2}-10 b^{6} c^{6} d \right ) x}{60 b^{7}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\) | \(459\) |
default | \(\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 b^{8} \left (b x +a \right )^{3}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{7 b^{8} \left (b x +a \right )^{7}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{5 b^{8} \left (b x +a \right )^{5}}+\frac {7 d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )}\) | \(462\) |
parallelrisch | \(\frac {2940 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}-70 a \,b^{6} c^{6} d -84 a^{2} b^{5} c^{5} d^{2}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-60 b^{7} c^{7}+1089 a^{7} d^{7}+8820 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}-2940 x^{2} a^{2} b^{5} c^{3} d^{4}-2205 x^{2} a \,b^{6} c^{4} d^{3}-14700 x^{3} a^{3} b^{4} c \,d^{6}-7350 x^{3} a^{2} b^{5} c^{2} d^{5}-4900 x^{3} a \,b^{6} c^{3} d^{4}-14700 x^{4} a^{2} b^{5} c \,d^{6}-7350 x^{4} a \,b^{6} c^{2} d^{5}-8820 x^{5} a \,b^{6} c \,d^{6}-2940 x \,a^{5} b^{2} c \,d^{6}-1470 x \,a^{4} b^{3} c^{2} d^{5}-980 x \,a^{3} b^{4} c^{3} d^{4}-735 x \,a^{2} b^{5} c^{4} d^{3}-588 x a \,b^{6} c^{5} d^{2}-8820 x^{2} a^{4} b^{3} c \,d^{6}-4410 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (b x +a \right ) x^{7} b^{7} d^{7}+420 \ln \left (b x +a \right ) a^{7} d^{7}+14700 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}+8820 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} d^{7}+2940 \ln \left (b x +a \right ) x^{6} a \,b^{6} d^{7}+13230 x^{5} a^{2} b^{5} d^{7}-4410 x^{5} b^{7} c^{2} d^{5}+2940 x^{6} a \,b^{6} d^{7}+7203 x \,a^{6} b \,d^{7}-490 x \,b^{7} c^{6} d +20139 x^{2} a^{5} b^{2} d^{7}-1764 x^{2} b^{7} c^{5} d^{2}+30625 x^{3} a^{4} b^{3} d^{7}-3675 x^{3} b^{7} c^{4} d^{3}+26950 x^{4} a^{3} b^{4} d^{7}-4900 x^{4} b^{7} c^{3} d^{4}-2940 x^{6} b^{7} c \,d^{6}+14700 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}}{420 b^{8} \left (b x +a \right )^{7}}\) | \(632\) |
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Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (182) = 364\).
Time = 0.23 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.22 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x - 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, a b^{6} d^{7} x^{6} + 21 \, a^{2} b^{5} d^{7} x^{5} + 35 \, a^{3} b^{4} d^{7} x^{4} + 35 \, a^{4} b^{3} d^{7} x^{3} + 21 \, a^{5} b^{2} d^{7} x^{2} + 7 \, a^{6} b d^{7} x + a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (182) = 364\).
Time = 0.23 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.75 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} + \frac {d^{7} \log \left (b x + a\right )}{b^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (182) = 364\).
Time = 0.29 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.40 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\frac {d^{7} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {2940 \, {\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 4410 \, {\left (b^{6} c^{2} d^{5} + 2 \, a b^{5} c d^{6} - 3 \, a^{2} b^{4} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{6} c^{3} d^{4} + 3 \, a b^{5} c^{2} d^{5} + 6 \, a^{2} b^{4} c d^{6} - 11 \, a^{3} b^{3} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{6} c^{4} d^{3} + 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 12 \, a^{3} b^{3} c d^{6} - 25 \, a^{4} b^{2} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{6} c^{5} d^{2} + 15 \, a b^{5} c^{4} d^{3} + 20 \, a^{2} b^{4} c^{3} d^{4} + 30 \, a^{3} b^{3} c^{2} d^{5} + 60 \, a^{4} b^{2} c d^{6} - 137 \, a^{5} b d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{6} c^{6} d + 12 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} + 20 \, a^{3} b^{3} c^{3} d^{4} + 30 \, a^{4} b^{2} c^{2} d^{5} + 60 \, a^{5} b c d^{6} - 147 \, a^{6} d^{7}\right )} x + \frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7}}{b}}{420 \, {\left (b x + a\right )}^{7} b^{7}} \]
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Time = 0.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.38 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\frac {d^7\,\ln \left (a+b\,x\right )}{b^8}-\frac {x\,\left (-\frac {343\,a^6\,b\,d^7}{20}+7\,a^5\,b^2\,c\,d^6+\frac {7\,a^4\,b^3\,c^2\,d^5}{2}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{4}+\frac {7\,a\,b^6\,c^5\,d^2}{5}+\frac {7\,b^7\,c^6\,d}{6}\right )-x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (-\frac {875\,a^4\,b^3\,d^7}{12}+35\,a^3\,b^4\,c\,d^6+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+\frac {35\,a\,b^6\,c^3\,d^4}{3}+\frac {35\,b^7\,c^4\,d^3}{4}\right )+x^5\,\left (-\frac {63\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6+\frac {21\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (-\frac {959\,a^5\,b^2\,d^7}{20}+21\,a^4\,b^3\,c\,d^6+\frac {21\,a^3\,b^4\,c^2\,d^5}{2}+7\,a^2\,b^5\,c^3\,d^4+\frac {21\,a\,b^6\,c^4\,d^3}{4}+\frac {21\,b^7\,c^5\,d^2}{5}\right )-\frac {363\,a^7\,d^7}{140}+\frac {b^7\,c^7}{7}+x^4\,\left (-\frac {385\,a^3\,b^4\,d^7}{6}+35\,a^2\,b^5\,c\,d^6+\frac {35\,a\,b^6\,c^2\,d^5}{2}+\frac {35\,b^7\,c^3\,d^4}{3}\right )+\frac {a^2\,b^5\,c^5\,d^2}{5}+\frac {a^3\,b^4\,c^4\,d^3}{4}+\frac {a^4\,b^3\,c^3\,d^4}{3}+\frac {a^5\,b^2\,c^2\,d^5}{2}+\frac {a\,b^6\,c^6\,d}{6}+a^6\,b\,c\,d^6}{b^8\,{\left (a+b\,x\right )}^7} \]
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