Integrand size = 16, antiderivative size = 231 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{10} a^{10} A x^{10}+\frac {1}{11} a^9 (10 A b+a B) x^{11}+\frac {5}{12} a^8 b (9 A b+2 a B) x^{12}+\frac {15}{13} a^7 b^2 (8 A b+3 a B) x^{13}+\frac {15}{7} a^6 b^3 (7 A b+4 a B) x^{14}+\frac {14}{5} a^5 b^4 (6 A b+5 a B) x^{15}+\frac {21}{8} a^4 b^5 (5 A b+6 a B) x^{16}+\frac {30}{17} a^3 b^6 (4 A b+7 a B) x^{17}+\frac {5}{6} a^2 b^7 (3 A b+8 a B) x^{18}+\frac {5}{19} a b^8 (2 A b+9 a B) x^{19}+\frac {1}{20} b^9 (A b+10 a B) x^{20}+\frac {1}{21} b^{10} B x^{21} \]
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Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{10} a^{10} A x^{10}+\frac {1}{11} a^9 x^{11} (a B+10 A b)+\frac {5}{12} a^8 b x^{12} (2 a B+9 A b)+\frac {15}{13} a^7 b^2 x^{13} (3 a B+8 A b)+\frac {15}{7} a^6 b^3 x^{14} (4 a B+7 A b)+\frac {14}{5} a^5 b^4 x^{15} (5 a B+6 A b)+\frac {21}{8} a^4 b^5 x^{16} (6 a B+5 A b)+\frac {30}{17} a^3 b^6 x^{17} (7 a B+4 A b)+\frac {5}{6} a^2 b^7 x^{18} (8 a B+3 A b)+\frac {1}{20} b^9 x^{20} (10 a B+A b)+\frac {5}{19} a b^8 x^{19} (9 a B+2 A b)+\frac {1}{21} b^{10} B x^{21} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^{10} A x^9+a^9 (10 A b+a B) x^{10}+5 a^8 b (9 A b+2 a B) x^{11}+15 a^7 b^2 (8 A b+3 a B) x^{12}+30 a^6 b^3 (7 A b+4 a B) x^{13}+42 a^5 b^4 (6 A b+5 a B) x^{14}+42 a^4 b^5 (5 A b+6 a B) x^{15}+30 a^3 b^6 (4 A b+7 a B) x^{16}+15 a^2 b^7 (3 A b+8 a B) x^{17}+5 a b^8 (2 A b+9 a B) x^{18}+b^9 (A b+10 a B) x^{19}+b^{10} B x^{20}\right ) \, dx \\ & = \frac {1}{10} a^{10} A x^{10}+\frac {1}{11} a^9 (10 A b+a B) x^{11}+\frac {5}{12} a^8 b (9 A b+2 a B) x^{12}+\frac {15}{13} a^7 b^2 (8 A b+3 a B) x^{13}+\frac {15}{7} a^6 b^3 (7 A b+4 a B) x^{14}+\frac {14}{5} a^5 b^4 (6 A b+5 a B) x^{15}+\frac {21}{8} a^4 b^5 (5 A b+6 a B) x^{16}+\frac {30}{17} a^3 b^6 (4 A b+7 a B) x^{17}+\frac {5}{6} a^2 b^7 (3 A b+8 a B) x^{18}+\frac {5}{19} a b^8 (2 A b+9 a B) x^{19}+\frac {1}{20} b^9 (A b+10 a B) x^{20}+\frac {1}{21} b^{10} B x^{21} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{10} a^{10} A x^{10}+\frac {1}{11} a^9 (10 A b+a B) x^{11}+\frac {5}{12} a^8 b (9 A b+2 a B) x^{12}+\frac {15}{13} a^7 b^2 (8 A b+3 a B) x^{13}+\frac {15}{7} a^6 b^3 (7 A b+4 a B) x^{14}+\frac {14}{5} a^5 b^4 (6 A b+5 a B) x^{15}+\frac {21}{8} a^4 b^5 (5 A b+6 a B) x^{16}+\frac {30}{17} a^3 b^6 (4 A b+7 a B) x^{17}+\frac {5}{6} a^2 b^7 (3 A b+8 a B) x^{18}+\frac {5}{19} a b^8 (2 A b+9 a B) x^{19}+\frac {1}{20} b^9 (A b+10 a B) x^{20}+\frac {1}{21} b^{10} B x^{21} \]
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Time = 0.40 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.02
method | result | size |
norman | \(\frac {a^{10} A \,x^{10}}{10}+\left (\frac {10}{11} a^{9} b A +\frac {1}{11} a^{10} B \right ) x^{11}+\left (\frac {15}{4} a^{8} b^{2} A +\frac {5}{6} a^{9} b B \right ) x^{12}+\left (\frac {120}{13} a^{7} b^{3} A +\frac {45}{13} a^{8} b^{2} B \right ) x^{13}+\left (15 a^{6} b^{4} A +\frac {60}{7} a^{7} b^{3} B \right ) x^{14}+\left (\frac {84}{5} a^{5} b^{5} A +14 a^{6} b^{4} B \right ) x^{15}+\left (\frac {105}{8} a^{4} b^{6} A +\frac {63}{4} a^{5} b^{5} B \right ) x^{16}+\left (\frac {120}{17} a^{3} b^{7} A +\frac {210}{17} a^{4} b^{6} B \right ) x^{17}+\left (\frac {5}{2} a^{2} b^{8} A +\frac {20}{3} a^{3} b^{7} B \right ) x^{18}+\left (\frac {10}{19} a \,b^{9} A +\frac {45}{19} a^{2} b^{8} B \right ) x^{19}+\left (\frac {1}{20} b^{10} A +\frac {1}{2} a \,b^{9} B \right ) x^{20}+\frac {b^{10} B \,x^{21}}{21}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{21}}{21}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{20}}{20}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{19}}{19}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{18}}{18}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{17}}{17}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{16}}{16}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{15}}{15}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{14}}{14}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{13}}{13}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{12}}{12}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{11}}{11}+\frac {a^{10} A \,x^{10}}{10}\) | \(244\) |
gosper | \(\frac {1}{10} a^{10} A \,x^{10}+\frac {10}{11} x^{11} a^{9} b A +\frac {1}{11} x^{11} a^{10} B +\frac {15}{4} x^{12} a^{8} b^{2} A +\frac {5}{6} x^{12} a^{9} b B +\frac {120}{13} x^{13} a^{7} b^{3} A +\frac {45}{13} x^{13} a^{8} b^{2} B +15 x^{14} a^{6} b^{4} A +\frac {60}{7} x^{14} a^{7} b^{3} B +\frac {84}{5} x^{15} a^{5} b^{5} A +14 x^{15} a^{6} b^{4} B +\frac {105}{8} x^{16} a^{4} b^{6} A +\frac {63}{4} x^{16} a^{5} b^{5} B +\frac {120}{17} x^{17} a^{3} b^{7} A +\frac {210}{17} x^{17} a^{4} b^{6} B +\frac {5}{2} x^{18} a^{2} b^{8} A +\frac {20}{3} x^{18} a^{3} b^{7} B +\frac {10}{19} x^{19} a \,b^{9} A +\frac {45}{19} x^{19} a^{2} b^{8} B +\frac {1}{20} x^{20} b^{10} A +\frac {1}{2} x^{20} a \,b^{9} B +\frac {1}{21} b^{10} B \,x^{21}\) | \(246\) |
risch | \(\frac {1}{10} a^{10} A \,x^{10}+\frac {10}{11} x^{11} a^{9} b A +\frac {1}{11} x^{11} a^{10} B +\frac {15}{4} x^{12} a^{8} b^{2} A +\frac {5}{6} x^{12} a^{9} b B +\frac {120}{13} x^{13} a^{7} b^{3} A +\frac {45}{13} x^{13} a^{8} b^{2} B +15 x^{14} a^{6} b^{4} A +\frac {60}{7} x^{14} a^{7} b^{3} B +\frac {84}{5} x^{15} a^{5} b^{5} A +14 x^{15} a^{6} b^{4} B +\frac {105}{8} x^{16} a^{4} b^{6} A +\frac {63}{4} x^{16} a^{5} b^{5} B +\frac {120}{17} x^{17} a^{3} b^{7} A +\frac {210}{17} x^{17} a^{4} b^{6} B +\frac {5}{2} x^{18} a^{2} b^{8} A +\frac {20}{3} x^{18} a^{3} b^{7} B +\frac {10}{19} x^{19} a \,b^{9} A +\frac {45}{19} x^{19} a^{2} b^{8} B +\frac {1}{20} x^{20} b^{10} A +\frac {1}{2} x^{20} a \,b^{9} B +\frac {1}{21} b^{10} B \,x^{21}\) | \(246\) |
parallelrisch | \(\frac {1}{10} a^{10} A \,x^{10}+\frac {10}{11} x^{11} a^{9} b A +\frac {1}{11} x^{11} a^{10} B +\frac {15}{4} x^{12} a^{8} b^{2} A +\frac {5}{6} x^{12} a^{9} b B +\frac {120}{13} x^{13} a^{7} b^{3} A +\frac {45}{13} x^{13} a^{8} b^{2} B +15 x^{14} a^{6} b^{4} A +\frac {60}{7} x^{14} a^{7} b^{3} B +\frac {84}{5} x^{15} a^{5} b^{5} A +14 x^{15} a^{6} b^{4} B +\frac {105}{8} x^{16} a^{4} b^{6} A +\frac {63}{4} x^{16} a^{5} b^{5} B +\frac {120}{17} x^{17} a^{3} b^{7} A +\frac {210}{17} x^{17} a^{4} b^{6} B +\frac {5}{2} x^{18} a^{2} b^{8} A +\frac {20}{3} x^{18} a^{3} b^{7} B +\frac {10}{19} x^{19} a \,b^{9} A +\frac {45}{19} x^{19} a^{2} b^{8} B +\frac {1}{20} x^{20} b^{10} A +\frac {1}{2} x^{20} a \,b^{9} B +\frac {1}{21} b^{10} B \,x^{21}\) | \(246\) |
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Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.05 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{21} \, B b^{10} x^{21} + \frac {1}{10} \, A a^{10} x^{10} + \frac {1}{20} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{20} + \frac {5}{19} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{19} + \frac {5}{6} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{18} + \frac {30}{17} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{17} + \frac {21}{8} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{16} + \frac {14}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{15} + \frac {15}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{14} + \frac {15}{13} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{13} + \frac {5}{12} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{12} + \frac {1}{11} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{11} \]
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Time = 0.04 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.16 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{10}}{10} + \frac {B b^{10} x^{21}}{21} + x^{20} \left (\frac {A b^{10}}{20} + \frac {B a b^{9}}{2}\right ) + x^{19} \cdot \left (\frac {10 A a b^{9}}{19} + \frac {45 B a^{2} b^{8}}{19}\right ) + x^{18} \cdot \left (\frac {5 A a^{2} b^{8}}{2} + \frac {20 B a^{3} b^{7}}{3}\right ) + x^{17} \cdot \left (\frac {120 A a^{3} b^{7}}{17} + \frac {210 B a^{4} b^{6}}{17}\right ) + x^{16} \cdot \left (\frac {105 A a^{4} b^{6}}{8} + \frac {63 B a^{5} b^{5}}{4}\right ) + x^{15} \cdot \left (\frac {84 A a^{5} b^{5}}{5} + 14 B a^{6} b^{4}\right ) + x^{14} \cdot \left (15 A a^{6} b^{4} + \frac {60 B a^{7} b^{3}}{7}\right ) + x^{13} \cdot \left (\frac {120 A a^{7} b^{3}}{13} + \frac {45 B a^{8} b^{2}}{13}\right ) + x^{12} \cdot \left (\frac {15 A a^{8} b^{2}}{4} + \frac {5 B a^{9} b}{6}\right ) + x^{11} \cdot \left (\frac {10 A a^{9} b}{11} + \frac {B a^{10}}{11}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.05 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{21} \, B b^{10} x^{21} + \frac {1}{10} \, A a^{10} x^{10} + \frac {1}{20} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{20} + \frac {5}{19} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{19} + \frac {5}{6} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{18} + \frac {30}{17} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{17} + \frac {21}{8} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{16} + \frac {14}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{15} + \frac {15}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{14} + \frac {15}{13} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{13} + \frac {5}{12} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{12} + \frac {1}{11} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{11} \]
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Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.06 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=\frac {1}{21} \, B b^{10} x^{21} + \frac {1}{2} \, B a b^{9} x^{20} + \frac {1}{20} \, A b^{10} x^{20} + \frac {45}{19} \, B a^{2} b^{8} x^{19} + \frac {10}{19} \, A a b^{9} x^{19} + \frac {20}{3} \, B a^{3} b^{7} x^{18} + \frac {5}{2} \, A a^{2} b^{8} x^{18} + \frac {210}{17} \, B a^{4} b^{6} x^{17} + \frac {120}{17} \, A a^{3} b^{7} x^{17} + \frac {63}{4} \, B a^{5} b^{5} x^{16} + \frac {105}{8} \, A a^{4} b^{6} x^{16} + 14 \, B a^{6} b^{4} x^{15} + \frac {84}{5} \, A a^{5} b^{5} x^{15} + \frac {60}{7} \, B a^{7} b^{3} x^{14} + 15 \, A a^{6} b^{4} x^{14} + \frac {45}{13} \, B a^{8} b^{2} x^{13} + \frac {120}{13} \, A a^{7} b^{3} x^{13} + \frac {5}{6} \, B a^{9} b x^{12} + \frac {15}{4} \, A a^{8} b^{2} x^{12} + \frac {1}{11} \, B a^{10} x^{11} + \frac {10}{11} \, A a^{9} b x^{11} + \frac {1}{10} \, A a^{10} x^{10} \]
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Time = 0.37 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.91 \[ \int x^9 (a+b x)^{10} (A+B x) \, dx=x^{11}\,\left (\frac {B\,a^{10}}{11}+\frac {10\,A\,b\,a^9}{11}\right )+x^{20}\,\left (\frac {A\,b^{10}}{20}+\frac {B\,a\,b^9}{2}\right )+\frac {A\,a^{10}\,x^{10}}{10}+\frac {B\,b^{10}\,x^{21}}{21}+\frac {15\,a^7\,b^2\,x^{13}\,\left (8\,A\,b+3\,B\,a\right )}{13}+\frac {15\,a^6\,b^3\,x^{14}\,\left (7\,A\,b+4\,B\,a\right )}{7}+\frac {14\,a^5\,b^4\,x^{15}\,\left (6\,A\,b+5\,B\,a\right )}{5}+\frac {21\,a^4\,b^5\,x^{16}\,\left (5\,A\,b+6\,B\,a\right )}{8}+\frac {30\,a^3\,b^6\,x^{17}\,\left (4\,A\,b+7\,B\,a\right )}{17}+\frac {5\,a^2\,b^7\,x^{18}\,\left (3\,A\,b+8\,B\,a\right )}{6}+\frac {5\,a^8\,b\,x^{12}\,\left (9\,A\,b+2\,B\,a\right )}{12}+\frac {5\,a\,b^8\,x^{19}\,\left (2\,A\,b+9\,B\,a\right )}{19} \]
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