Integrand size = 14, antiderivative size = 62 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {c (b c-a d) (c+d x)^{17}}{17 d^3}-\frac {(2 b c-a d) (c+d x)^{18}}{18 d^3}+\frac {b (c+d x)^{19}}{19 d^3} \]
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Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ \int x (a+b x) (c+d x)^{16} \, dx=-\frac {(c+d x)^{18} (2 b c-a d)}{18 d^3}+\frac {c (c+d x)^{17} (b c-a d)}{17 d^3}+\frac {b (c+d x)^{19}}{19 d^3} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c (b c-a d) (c+d x)^{16}}{d^2}+\frac {(-2 b c+a d) (c+d x)^{17}}{d^2}+\frac {b (c+d x)^{18}}{d^2}\right ) \, dx \\ & = \frac {c (b c-a d) (c+d x)^{17}}{17 d^3}-\frac {(2 b c-a d) (c+d x)^{18}}{18 d^3}+\frac {b (c+d x)^{19}}{19 d^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(347\) vs. \(2(62)=124\).
Time = 0.03 (sec) , antiderivative size = 347, normalized size of antiderivative = 5.60 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {1}{2} a c^{16} x^2+\frac {1}{3} c^{15} (b c+16 a d) x^3+2 c^{14} d (2 b c+15 a d) x^4+8 c^{13} d^2 (3 b c+14 a d) x^5+\frac {70}{3} c^{12} d^3 (4 b c+13 a d) x^6+52 c^{11} d^4 (5 b c+12 a d) x^7+91 c^{10} d^5 (6 b c+11 a d) x^8+\frac {1144}{9} c^9 d^6 (7 b c+10 a d) x^9+143 c^8 d^7 (8 b c+9 a d) x^{10}+130 c^7 d^8 (9 b c+8 a d) x^{11}+\frac {286}{3} c^6 d^9 (10 b c+7 a d) x^{12}+56 c^5 d^{10} (11 b c+6 a d) x^{13}+26 c^4 d^{11} (12 b c+5 a d) x^{14}+\frac {28}{3} c^3 d^{12} (13 b c+4 a d) x^{15}+\frac {5}{2} c^2 d^{13} (14 b c+3 a d) x^{16}+\frac {8}{17} c d^{14} (15 b c+2 a d) x^{17}+\frac {1}{18} d^{15} (16 b c+a d) x^{18}+\frac {1}{19} b d^{16} x^{19} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(56)=112\).
Time = 0.40 (sec) , antiderivative size = 374, normalized size of antiderivative = 6.03
method | result | size |
norman | \(\frac {a \,c^{16} x^{2}}{2}+\left (\frac {16}{3} a \,c^{15} d +\frac {1}{3} b \,c^{16}\right ) x^{3}+\left (30 a \,c^{14} d^{2}+4 b \,c^{15} d \right ) x^{4}+\left (112 a \,c^{13} d^{3}+24 b \,c^{14} d^{2}\right ) x^{5}+\left (\frac {910}{3} a \,c^{12} d^{4}+\frac {280}{3} b \,c^{13} d^{3}\right ) x^{6}+\left (624 a \,c^{11} d^{5}+260 b \,c^{12} d^{4}\right ) x^{7}+\left (1001 a \,c^{10} d^{6}+546 b \,c^{11} d^{5}\right ) x^{8}+\left (\frac {11440}{9} a \,c^{9} d^{7}+\frac {8008}{9} b \,c^{10} d^{6}\right ) x^{9}+\left (1287 a \,c^{8} d^{8}+1144 b \,c^{9} d^{7}\right ) x^{10}+\left (1040 a \,c^{7} d^{9}+1170 b \,c^{8} d^{8}\right ) x^{11}+\left (\frac {2002}{3} a \,c^{6} d^{10}+\frac {2860}{3} b \,c^{7} d^{9}\right ) x^{12}+\left (336 a \,c^{5} d^{11}+616 b \,c^{6} d^{10}\right ) x^{13}+\left (130 a \,c^{4} d^{12}+312 b \,c^{5} d^{11}\right ) x^{14}+\left (\frac {112}{3} a \,c^{3} d^{13}+\frac {364}{3} b \,c^{4} d^{12}\right ) x^{15}+\left (\frac {15}{2} a \,c^{2} d^{14}+35 b \,c^{3} d^{13}\right ) x^{16}+\left (\frac {16}{17} a c \,d^{15}+\frac {120}{17} b \,c^{2} d^{14}\right ) x^{17}+\left (\frac {1}{18} a \,d^{16}+\frac {8}{9} b c \,d^{15}\right ) x^{18}+\frac {b \,d^{16} x^{19}}{19}\) | \(374\) |
default | \(\frac {b \,d^{16} x^{19}}{19}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{18}}{18}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{17}}{17}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{16}}{16}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{15}}{15}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{14}}{14}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{13}}{13}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{12}}{12}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{11}}{11}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{10}}{10}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{9}}{9}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{8}}{8}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{7}}{7}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{6}}{6}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{5}}{5}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{4}}{4}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{3}}{3}+\frac {a \,c^{16} x^{2}}{2}\) | \(388\) |
gosper | \(\frac {1}{19} b \,d^{16} x^{19}+\frac {1}{2} a \,c^{16} x^{2}+\frac {1}{3} x^{3} b \,c^{16}+\frac {1}{18} x^{18} a \,d^{16}+1040 a \,c^{7} d^{9} x^{11}+1170 b \,c^{8} d^{8} x^{11}+336 a \,c^{5} d^{11} x^{13}+616 b \,c^{6} d^{10} x^{13}+130 a \,c^{4} d^{12} x^{14}+312 b \,c^{5} d^{11} x^{14}+\frac {112}{3} x^{15} a \,c^{3} d^{13}+\frac {364}{3} x^{15} b \,c^{4} d^{12}+\frac {15}{2} x^{16} a \,c^{2} d^{14}+35 x^{16} b \,c^{3} d^{13}+\frac {16}{17} x^{17} a c \,d^{15}+\frac {120}{17} x^{17} b \,c^{2} d^{14}+\frac {8}{9} x^{18} b c \,d^{15}+30 a \,c^{14} d^{2} x^{4}+4 b \,c^{15} d \,x^{4}+112 a \,c^{13} d^{3} x^{5}+24 b \,c^{14} d^{2} x^{5}+624 a \,c^{11} d^{5} x^{7}+260 b \,c^{12} d^{4} x^{7}+1001 a \,c^{10} d^{6} x^{8}+546 b \,c^{11} d^{5} x^{8}+1287 a \,c^{8} d^{8} x^{10}+1144 b \,c^{9} d^{7} x^{10}+\frac {16}{3} x^{3} a \,c^{15} d +\frac {910}{3} x^{6} a \,c^{12} d^{4}+\frac {280}{3} x^{6} b \,c^{13} d^{3}+\frac {11440}{9} x^{9} a \,c^{9} d^{7}+\frac {8008}{9} x^{9} b \,c^{10} d^{6}+\frac {2002}{3} x^{12} a \,c^{6} d^{10}+\frac {2860}{3} x^{12} b \,c^{7} d^{9}\) | \(390\) |
risch | \(\frac {1}{19} b \,d^{16} x^{19}+\frac {1}{2} a \,c^{16} x^{2}+\frac {1}{3} x^{3} b \,c^{16}+\frac {1}{18} x^{18} a \,d^{16}+1040 a \,c^{7} d^{9} x^{11}+1170 b \,c^{8} d^{8} x^{11}+336 a \,c^{5} d^{11} x^{13}+616 b \,c^{6} d^{10} x^{13}+130 a \,c^{4} d^{12} x^{14}+312 b \,c^{5} d^{11} x^{14}+\frac {112}{3} x^{15} a \,c^{3} d^{13}+\frac {364}{3} x^{15} b \,c^{4} d^{12}+\frac {15}{2} x^{16} a \,c^{2} d^{14}+35 x^{16} b \,c^{3} d^{13}+\frac {16}{17} x^{17} a c \,d^{15}+\frac {120}{17} x^{17} b \,c^{2} d^{14}+\frac {8}{9} x^{18} b c \,d^{15}+30 a \,c^{14} d^{2} x^{4}+4 b \,c^{15} d \,x^{4}+112 a \,c^{13} d^{3} x^{5}+24 b \,c^{14} d^{2} x^{5}+624 a \,c^{11} d^{5} x^{7}+260 b \,c^{12} d^{4} x^{7}+1001 a \,c^{10} d^{6} x^{8}+546 b \,c^{11} d^{5} x^{8}+1287 a \,c^{8} d^{8} x^{10}+1144 b \,c^{9} d^{7} x^{10}+\frac {16}{3} x^{3} a \,c^{15} d +\frac {910}{3} x^{6} a \,c^{12} d^{4}+\frac {280}{3} x^{6} b \,c^{13} d^{3}+\frac {11440}{9} x^{9} a \,c^{9} d^{7}+\frac {8008}{9} x^{9} b \,c^{10} d^{6}+\frac {2002}{3} x^{12} a \,c^{6} d^{10}+\frac {2860}{3} x^{12} b \,c^{7} d^{9}\) | \(390\) |
parallelrisch | \(\frac {1}{19} b \,d^{16} x^{19}+\frac {1}{2} a \,c^{16} x^{2}+\frac {1}{3} x^{3} b \,c^{16}+\frac {1}{18} x^{18} a \,d^{16}+1040 a \,c^{7} d^{9} x^{11}+1170 b \,c^{8} d^{8} x^{11}+336 a \,c^{5} d^{11} x^{13}+616 b \,c^{6} d^{10} x^{13}+130 a \,c^{4} d^{12} x^{14}+312 b \,c^{5} d^{11} x^{14}+\frac {112}{3} x^{15} a \,c^{3} d^{13}+\frac {364}{3} x^{15} b \,c^{4} d^{12}+\frac {15}{2} x^{16} a \,c^{2} d^{14}+35 x^{16} b \,c^{3} d^{13}+\frac {16}{17} x^{17} a c \,d^{15}+\frac {120}{17} x^{17} b \,c^{2} d^{14}+\frac {8}{9} x^{18} b c \,d^{15}+30 a \,c^{14} d^{2} x^{4}+4 b \,c^{15} d \,x^{4}+112 a \,c^{13} d^{3} x^{5}+24 b \,c^{14} d^{2} x^{5}+624 a \,c^{11} d^{5} x^{7}+260 b \,c^{12} d^{4} x^{7}+1001 a \,c^{10} d^{6} x^{8}+546 b \,c^{11} d^{5} x^{8}+1287 a \,c^{8} d^{8} x^{10}+1144 b \,c^{9} d^{7} x^{10}+\frac {16}{3} x^{3} a \,c^{15} d +\frac {910}{3} x^{6} a \,c^{12} d^{4}+\frac {280}{3} x^{6} b \,c^{13} d^{3}+\frac {11440}{9} x^{9} a \,c^{9} d^{7}+\frac {8008}{9} x^{9} b \,c^{10} d^{6}+\frac {2002}{3} x^{12} a \,c^{6} d^{10}+\frac {2860}{3} x^{12} b \,c^{7} d^{9}\) | \(390\) |
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (56) = 112\).
Time = 0.22 (sec) , antiderivative size = 387, normalized size of antiderivative = 6.24 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {1}{19} \, b d^{16} x^{19} + \frac {1}{2} \, a c^{16} x^{2} + \frac {1}{18} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{18} + \frac {8}{17} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{17} + \frac {5}{2} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{16} + \frac {28}{3} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{15} + 26 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{14} + 56 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{13} + \frac {286}{3} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{12} + 130 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{11} + 143 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{10} + \frac {1144}{9} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{9} + 91 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{8} + 52 \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{7} + \frac {70}{3} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{6} + 8 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{5} + 2 \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (53) = 106\).
Time = 0.06 (sec) , antiderivative size = 408, normalized size of antiderivative = 6.58 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {a c^{16} x^{2}}{2} + \frac {b d^{16} x^{19}}{19} + x^{18} \left (\frac {a d^{16}}{18} + \frac {8 b c d^{15}}{9}\right ) + x^{17} \cdot \left (\frac {16 a c d^{15}}{17} + \frac {120 b c^{2} d^{14}}{17}\right ) + x^{16} \cdot \left (\frac {15 a c^{2} d^{14}}{2} + 35 b c^{3} d^{13}\right ) + x^{15} \cdot \left (\frac {112 a c^{3} d^{13}}{3} + \frac {364 b c^{4} d^{12}}{3}\right ) + x^{14} \cdot \left (130 a c^{4} d^{12} + 312 b c^{5} d^{11}\right ) + x^{13} \cdot \left (336 a c^{5} d^{11} + 616 b c^{6} d^{10}\right ) + x^{12} \cdot \left (\frac {2002 a c^{6} d^{10}}{3} + \frac {2860 b c^{7} d^{9}}{3}\right ) + x^{11} \cdot \left (1040 a c^{7} d^{9} + 1170 b c^{8} d^{8}\right ) + x^{10} \cdot \left (1287 a c^{8} d^{8} + 1144 b c^{9} d^{7}\right ) + x^{9} \cdot \left (\frac {11440 a c^{9} d^{7}}{9} + \frac {8008 b c^{10} d^{6}}{9}\right ) + x^{8} \cdot \left (1001 a c^{10} d^{6} + 546 b c^{11} d^{5}\right ) + x^{7} \cdot \left (624 a c^{11} d^{5} + 260 b c^{12} d^{4}\right ) + x^{6} \cdot \left (\frac {910 a c^{12} d^{4}}{3} + \frac {280 b c^{13} d^{3}}{3}\right ) + x^{5} \cdot \left (112 a c^{13} d^{3} + 24 b c^{14} d^{2}\right ) + x^{4} \cdot \left (30 a c^{14} d^{2} + 4 b c^{15} d\right ) + x^{3} \cdot \left (\frac {16 a c^{15} d}{3} + \frac {b c^{16}}{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (56) = 112\).
Time = 0.19 (sec) , antiderivative size = 387, normalized size of antiderivative = 6.24 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {1}{19} \, b d^{16} x^{19} + \frac {1}{2} \, a c^{16} x^{2} + \frac {1}{18} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{18} + \frac {8}{17} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{17} + \frac {5}{2} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{16} + \frac {28}{3} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{15} + 26 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{14} + 56 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{13} + \frac {286}{3} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{12} + 130 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{11} + 143 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{10} + \frac {1144}{9} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{9} + 91 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{8} + 52 \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{7} + \frac {70}{3} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{6} + 8 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{5} + 2 \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 389, normalized size of antiderivative = 6.27 \[ \int x (a+b x) (c+d x)^{16} \, dx=\frac {1}{19} \, b d^{16} x^{19} + \frac {8}{9} \, b c d^{15} x^{18} + \frac {1}{18} \, a d^{16} x^{18} + \frac {120}{17} \, b c^{2} d^{14} x^{17} + \frac {16}{17} \, a c d^{15} x^{17} + 35 \, b c^{3} d^{13} x^{16} + \frac {15}{2} \, a c^{2} d^{14} x^{16} + \frac {364}{3} \, b c^{4} d^{12} x^{15} + \frac {112}{3} \, a c^{3} d^{13} x^{15} + 312 \, b c^{5} d^{11} x^{14} + 130 \, a c^{4} d^{12} x^{14} + 616 \, b c^{6} d^{10} x^{13} + 336 \, a c^{5} d^{11} x^{13} + \frac {2860}{3} \, b c^{7} d^{9} x^{12} + \frac {2002}{3} \, a c^{6} d^{10} x^{12} + 1170 \, b c^{8} d^{8} x^{11} + 1040 \, a c^{7} d^{9} x^{11} + 1144 \, b c^{9} d^{7} x^{10} + 1287 \, a c^{8} d^{8} x^{10} + \frac {8008}{9} \, b c^{10} d^{6} x^{9} + \frac {11440}{9} \, a c^{9} d^{7} x^{9} + 546 \, b c^{11} d^{5} x^{8} + 1001 \, a c^{10} d^{6} x^{8} + 260 \, b c^{12} d^{4} x^{7} + 624 \, a c^{11} d^{5} x^{7} + \frac {280}{3} \, b c^{13} d^{3} x^{6} + \frac {910}{3} \, a c^{12} d^{4} x^{6} + 24 \, b c^{14} d^{2} x^{5} + 112 \, a c^{13} d^{3} x^{5} + 4 \, b c^{15} d x^{4} + 30 \, a c^{14} d^{2} x^{4} + \frac {1}{3} \, b c^{16} x^{3} + \frac {16}{3} \, a c^{15} d x^{3} + \frac {1}{2} \, a c^{16} x^{2} \]
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Time = 0.26 (sec) , antiderivative size = 331, normalized size of antiderivative = 5.34 \[ \int x (a+b x) (c+d x)^{16} \, dx=x^3\,\left (\frac {b\,c^{16}}{3}+\frac {16\,a\,d\,c^{15}}{3}\right )+x^{18}\,\left (\frac {a\,d^{16}}{18}+\frac {8\,b\,c\,d^{15}}{9}\right )+\frac {a\,c^{16}\,x^2}{2}+\frac {b\,d^{16}\,x^{19}}{19}+2\,c^{14}\,d\,x^4\,\left (15\,a\,d+2\,b\,c\right )+\frac {8\,c\,d^{14}\,x^{17}\,\left (2\,a\,d+15\,b\,c\right )}{17}+8\,c^{13}\,d^2\,x^5\,\left (14\,a\,d+3\,b\,c\right )+\frac {70\,c^{12}\,d^3\,x^6\,\left (13\,a\,d+4\,b\,c\right )}{3}+52\,c^{11}\,d^4\,x^7\,\left (12\,a\,d+5\,b\,c\right )+91\,c^{10}\,d^5\,x^8\,\left (11\,a\,d+6\,b\,c\right )+\frac {1144\,c^9\,d^6\,x^9\,\left (10\,a\,d+7\,b\,c\right )}{9}+143\,c^8\,d^7\,x^{10}\,\left (9\,a\,d+8\,b\,c\right )+130\,c^7\,d^8\,x^{11}\,\left (8\,a\,d+9\,b\,c\right )+\frac {286\,c^6\,d^9\,x^{12}\,\left (7\,a\,d+10\,b\,c\right )}{3}+56\,c^5\,d^{10}\,x^{13}\,\left (6\,a\,d+11\,b\,c\right )+26\,c^4\,d^{11}\,x^{14}\,\left (5\,a\,d+12\,b\,c\right )+\frac {28\,c^3\,d^{12}\,x^{15}\,\left (4\,a\,d+13\,b\,c\right )}{3}+\frac {5\,c^2\,d^{13}\,x^{16}\,\left (3\,a\,d+14\,b\,c\right )}{2} \]
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