Integrand size = 13, antiderivative size = 38 \[ \int (a+b x) (c+d x)^{16} \, dx=-\frac {(b c-a d) (c+d x)^{17}}{17 d^2}+\frac {b (c+d x)^{18}}{18 d^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {b (c+d x)^{18}}{18 d^2}-\frac {(c+d x)^{17} (b c-a d)}{17 d^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d) (c+d x)^{16}}{d}+\frac {b (c+d x)^{17}}{d}\right ) \, dx \\ & = -\frac {(b c-a d) (c+d x)^{17}}{17 d^2}+\frac {b (c+d x)^{18}}{18 d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(38)=76\).
Time = 0.03 (sec) , antiderivative size = 342, normalized size of antiderivative = 9.00 \[ \int (a+b x) (c+d x)^{16} \, dx=a c^{16} x+\frac {1}{2} c^{15} (b c+16 a d) x^2+\frac {8}{3} c^{14} d (2 b c+15 a d) x^3+10 c^{13} d^2 (3 b c+14 a d) x^4+28 c^{12} d^3 (4 b c+13 a d) x^5+\frac {182}{3} c^{11} d^4 (5 b c+12 a d) x^6+104 c^{10} d^5 (6 b c+11 a d) x^7+143 c^9 d^6 (7 b c+10 a d) x^8+\frac {1430}{9} c^8 d^7 (8 b c+9 a d) x^9+143 c^7 d^8 (9 b c+8 a d) x^{10}+104 c^6 d^9 (10 b c+7 a d) x^{11}+\frac {182}{3} c^5 d^{10} (11 b c+6 a d) x^{12}+28 c^4 d^{11} (12 b c+5 a d) x^{13}+10 c^3 d^{12} (13 b c+4 a d) x^{14}+\frac {8}{3} c^2 d^{13} (14 b c+3 a d) x^{15}+\frac {1}{2} c d^{14} (15 b c+2 a d) x^{16}+\frac {1}{17} d^{15} (16 b c+a d) x^{17}+\frac {1}{18} b d^{16} x^{18} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(34)=68\).
Time = 0.41 (sec) , antiderivative size = 370, normalized size of antiderivative = 9.74
method | result | size |
norman | \(a \,c^{16} x +\left (8 a \,c^{15} d +\frac {1}{2} b \,c^{16}\right ) x^{2}+\left (40 a \,c^{14} d^{2}+\frac {16}{3} b \,c^{15} d \right ) x^{3}+\left (140 a \,c^{13} d^{3}+30 b \,c^{14} d^{2}\right ) x^{4}+\left (364 a \,c^{12} d^{4}+112 b \,c^{13} d^{3}\right ) x^{5}+\left (728 a \,c^{11} d^{5}+\frac {910}{3} b \,c^{12} d^{4}\right ) x^{6}+\left (1144 a \,c^{10} d^{6}+624 b \,c^{11} d^{5}\right ) x^{7}+\left (1430 a \,c^{9} d^{7}+1001 b \,c^{10} d^{6}\right ) x^{8}+\left (1430 a \,c^{8} d^{8}+\frac {11440}{9} b \,c^{9} d^{7}\right ) x^{9}+\left (1144 a \,c^{7} d^{9}+1287 b \,c^{8} d^{8}\right ) x^{10}+\left (728 a \,c^{6} d^{10}+1040 b \,c^{7} d^{9}\right ) x^{11}+\left (364 a \,c^{5} d^{11}+\frac {2002}{3} b \,c^{6} d^{10}\right ) x^{12}+\left (140 a \,c^{4} d^{12}+336 b \,c^{5} d^{11}\right ) x^{13}+\left (40 a \,c^{3} d^{13}+130 b \,c^{4} d^{12}\right ) x^{14}+\left (8 a \,c^{2} d^{14}+\frac {112}{3} b \,c^{3} d^{13}\right ) x^{15}+\left (a c \,d^{15}+\frac {15}{2} b \,c^{2} d^{14}\right ) x^{16}+\left (\frac {1}{17} a \,d^{16}+\frac {16}{17} b c \,d^{15}\right ) x^{17}+\frac {b \,d^{16} x^{18}}{18}\) | \(370\) |
default | \(\frac {b \,d^{16} x^{18}}{18}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{17}}{17}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{16}}{16}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{15}}{15}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{14}}{14}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{13}}{13}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{12}}{12}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{11}}{11}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{10}}{10}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{9}}{9}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{8}}{8}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{7}}{7}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{6}}{6}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{5}}{5}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{4}}{4}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{3}}{3}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{2}}{2}+a \,c^{16} x\) | \(385\) |
gosper | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
risch | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
parallelrisch | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 10.11 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + a c^{16} x + \frac {1}{17} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{16} + \frac {8}{3} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{15} + 10 \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{14} + 28 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{13} + \frac {182}{3} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{12} + 104 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{11} + 143 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{10} + \frac {1430}{9} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{9} + 143 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{8} + 104 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{7} + \frac {182}{3} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{6} + 28 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{5} + 10 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{4} + \frac {8}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (32) = 64\).
Time = 0.07 (sec) , antiderivative size = 393, normalized size of antiderivative = 10.34 \[ \int (a+b x) (c+d x)^{16} \, dx=a c^{16} x + \frac {b d^{16} x^{18}}{18} + x^{17} \left (\frac {a d^{16}}{17} + \frac {16 b c d^{15}}{17}\right ) + x^{16} \left (a c d^{15} + \frac {15 b c^{2} d^{14}}{2}\right ) + x^{15} \cdot \left (8 a c^{2} d^{14} + \frac {112 b c^{3} d^{13}}{3}\right ) + x^{14} \cdot \left (40 a c^{3} d^{13} + 130 b c^{4} d^{12}\right ) + x^{13} \cdot \left (140 a c^{4} d^{12} + 336 b c^{5} d^{11}\right ) + x^{12} \cdot \left (364 a c^{5} d^{11} + \frac {2002 b c^{6} d^{10}}{3}\right ) + x^{11} \cdot \left (728 a c^{6} d^{10} + 1040 b c^{7} d^{9}\right ) + x^{10} \cdot \left (1144 a c^{7} d^{9} + 1287 b c^{8} d^{8}\right ) + x^{9} \cdot \left (1430 a c^{8} d^{8} + \frac {11440 b c^{9} d^{7}}{9}\right ) + x^{8} \cdot \left (1430 a c^{9} d^{7} + 1001 b c^{10} d^{6}\right ) + x^{7} \cdot \left (1144 a c^{10} d^{6} + 624 b c^{11} d^{5}\right ) + x^{6} \cdot \left (728 a c^{11} d^{5} + \frac {910 b c^{12} d^{4}}{3}\right ) + x^{5} \cdot \left (364 a c^{12} d^{4} + 112 b c^{13} d^{3}\right ) + x^{4} \cdot \left (140 a c^{13} d^{3} + 30 b c^{14} d^{2}\right ) + x^{3} \cdot \left (40 a c^{14} d^{2} + \frac {16 b c^{15} d}{3}\right ) + x^{2} \cdot \left (8 a c^{15} d + \frac {b c^{16}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 384, normalized size of antiderivative = 10.11 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + a c^{16} x + \frac {1}{17} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{16} + \frac {8}{3} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{15} + 10 \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{14} + 28 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{13} + \frac {182}{3} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{12} + 104 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{11} + 143 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{10} + \frac {1430}{9} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{9} + 143 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{8} + 104 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{7} + \frac {182}{3} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{6} + 28 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{5} + 10 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{4} + \frac {8}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 10.13 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + \frac {16}{17} \, b c d^{15} x^{17} + \frac {1}{17} \, a d^{16} x^{17} + \frac {15}{2} \, b c^{2} d^{14} x^{16} + a c d^{15} x^{16} + \frac {112}{3} \, b c^{3} d^{13} x^{15} + 8 \, a c^{2} d^{14} x^{15} + 130 \, b c^{4} d^{12} x^{14} + 40 \, a c^{3} d^{13} x^{14} + 336 \, b c^{5} d^{11} x^{13} + 140 \, a c^{4} d^{12} x^{13} + \frac {2002}{3} \, b c^{6} d^{10} x^{12} + 364 \, a c^{5} d^{11} x^{12} + 1040 \, b c^{7} d^{9} x^{11} + 728 \, a c^{6} d^{10} x^{11} + 1287 \, b c^{8} d^{8} x^{10} + 1144 \, a c^{7} d^{9} x^{10} + \frac {11440}{9} \, b c^{9} d^{7} x^{9} + 1430 \, a c^{8} d^{8} x^{9} + 1001 \, b c^{10} d^{6} x^{8} + 1430 \, a c^{9} d^{7} x^{8} + 624 \, b c^{11} d^{5} x^{7} + 1144 \, a c^{10} d^{6} x^{7} + \frac {910}{3} \, b c^{12} d^{4} x^{6} + 728 \, a c^{11} d^{5} x^{6} + 112 \, b c^{13} d^{3} x^{5} + 364 \, a c^{12} d^{4} x^{5} + 30 \, b c^{14} d^{2} x^{4} + 140 \, a c^{13} d^{3} x^{4} + \frac {16}{3} \, b c^{15} d x^{3} + 40 \, a c^{14} d^{2} x^{3} + \frac {1}{2} \, b c^{16} x^{2} + 8 \, a c^{15} d x^{2} + a c^{16} x \]
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Time = 0.56 (sec) , antiderivative size = 328, normalized size of antiderivative = 8.63 \[ \int (a+b x) (c+d x)^{16} \, dx=x^2\,\left (\frac {b\,c^{16}}{2}+8\,a\,d\,c^{15}\right )+x^{17}\,\left (\frac {a\,d^{16}}{17}+\frac {16\,b\,c\,d^{15}}{17}\right )+\frac {b\,d^{16}\,x^{18}}{18}+a\,c^{16}\,x+\frac {8\,c^{14}\,d\,x^3\,\left (15\,a\,d+2\,b\,c\right )}{3}+\frac {c\,d^{14}\,x^{16}\,\left (2\,a\,d+15\,b\,c\right )}{2}+10\,c^{13}\,d^2\,x^4\,\left (14\,a\,d+3\,b\,c\right )+28\,c^{12}\,d^3\,x^5\,\left (13\,a\,d+4\,b\,c\right )+\frac {182\,c^{11}\,d^4\,x^6\,\left (12\,a\,d+5\,b\,c\right )}{3}+104\,c^{10}\,d^5\,x^7\,\left (11\,a\,d+6\,b\,c\right )+143\,c^9\,d^6\,x^8\,\left (10\,a\,d+7\,b\,c\right )+\frac {1430\,c^8\,d^7\,x^9\,\left (9\,a\,d+8\,b\,c\right )}{9}+143\,c^7\,d^8\,x^{10}\,\left (8\,a\,d+9\,b\,c\right )+104\,c^6\,d^9\,x^{11}\,\left (7\,a\,d+10\,b\,c\right )+\frac {182\,c^5\,d^{10}\,x^{12}\,\left (6\,a\,d+11\,b\,c\right )}{3}+28\,c^4\,d^{11}\,x^{13}\,\left (5\,a\,d+12\,b\,c\right )+10\,c^3\,d^{12}\,x^{14}\,\left (4\,a\,d+13\,b\,c\right )+\frac {8\,c^2\,d^{13}\,x^{15}\,\left (3\,a\,d+14\,b\,c\right )}{3} \]
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