Integrand size = 16, antiderivative size = 240 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {a^8 (A b-a B) (a+b x)^{11}}{11 b^{10}}-\frac {a^7 (8 A b-9 a B) (a+b x)^{12}}{12 b^{10}}+\frac {4 a^6 (7 A b-9 a B) (a+b x)^{13}}{13 b^{10}}-\frac {2 a^5 (2 A b-3 a B) (a+b x)^{14}}{b^{10}}+\frac {14 a^4 (5 A b-9 a B) (a+b x)^{15}}{15 b^{10}}-\frac {7 a^3 (4 A b-9 a B) (a+b x)^{16}}{8 b^{10}}+\frac {28 a^2 (A b-3 a B) (a+b x)^{17}}{17 b^{10}}-\frac {2 a (2 A b-9 a B) (a+b x)^{18}}{9 b^{10}}+\frac {(A b-9 a B) (a+b x)^{19}}{19 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}} \]
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Time = 0.13 (sec) , antiderivative size = 240, 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} \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}-\frac {a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac {4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac {2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac {14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac {7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac {28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}+\frac {(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac {2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^8 (-A b+a B) (a+b x)^{10}}{b^9}+\frac {a^7 (-8 A b+9 a B) (a+b x)^{11}}{b^9}-\frac {4 a^6 (-7 A b+9 a B) (a+b x)^{12}}{b^9}+\frac {28 a^5 (-2 A b+3 a B) (a+b x)^{13}}{b^9}-\frac {14 a^4 (-5 A b+9 a B) (a+b x)^{14}}{b^9}+\frac {14 a^3 (-4 A b+9 a B) (a+b x)^{15}}{b^9}-\frac {28 a^2 (-A b+3 a B) (a+b x)^{16}}{b^9}+\frac {4 a (-2 A b+9 a B) (a+b x)^{17}}{b^9}+\frac {(A b-9 a B) (a+b x)^{18}}{b^9}+\frac {B (a+b x)^{19}}{b^9}\right ) \, dx \\ & = \frac {a^8 (A b-a B) (a+b x)^{11}}{11 b^{10}}-\frac {a^7 (8 A b-9 a B) (a+b x)^{12}}{12 b^{10}}+\frac {4 a^6 (7 A b-9 a B) (a+b x)^{13}}{13 b^{10}}-\frac {2 a^5 (2 A b-3 a B) (a+b x)^{14}}{b^{10}}+\frac {14 a^4 (5 A b-9 a B) (a+b x)^{15}}{15 b^{10}}-\frac {7 a^3 (4 A b-9 a B) (a+b x)^{16}}{8 b^{10}}+\frac {28 a^2 (A b-3 a B) (a+b x)^{17}}{17 b^{10}}-\frac {2 a (2 A b-9 a B) (a+b x)^{18}}{9 b^{10}}+\frac {(A b-9 a B) (a+b x)^{19}}{19 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{9} a^{10} A x^9+\frac {1}{10} a^9 (10 A b+a B) x^{10}+\frac {5}{11} a^8 b (9 A b+2 a B) x^{11}+\frac {5}{4} a^7 b^2 (8 A b+3 a B) x^{12}+\frac {30}{13} a^6 b^3 (7 A b+4 a B) x^{13}+3 a^5 b^4 (6 A b+5 a B) x^{14}+\frac {14}{5} a^4 b^5 (5 A b+6 a B) x^{15}+\frac {15}{8} a^3 b^6 (4 A b+7 a B) x^{16}+\frac {15}{17} a^2 b^7 (3 A b+8 a B) x^{17}+\frac {5}{18} a b^8 (2 A b+9 a B) x^{18}+\frac {1}{19} b^9 (A b+10 a B) x^{19}+\frac {1}{20} b^{10} B x^{20} \]
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Time = 0.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {a^{10} A \,x^{9}}{9}+\left (a^{9} b A +\frac {1}{10} a^{10} B \right ) x^{10}+\left (\frac {45}{11} a^{8} b^{2} A +\frac {10}{11} a^{9} b B \right ) x^{11}+\left (10 a^{7} b^{3} A +\frac {15}{4} a^{8} b^{2} B \right ) x^{12}+\left (\frac {210}{13} a^{6} b^{4} A +\frac {120}{13} a^{7} b^{3} B \right ) x^{13}+\left (18 a^{5} b^{5} A +15 a^{6} b^{4} B \right ) x^{14}+\left (14 a^{4} b^{6} A +\frac {84}{5} a^{5} b^{5} B \right ) x^{15}+\left (\frac {15}{2} a^{3} b^{7} A +\frac {105}{8} a^{4} b^{6} B \right ) x^{16}+\left (\frac {45}{17} a^{2} b^{8} A +\frac {120}{17} a^{3} b^{7} B \right ) x^{17}+\left (\frac {5}{9} a \,b^{9} A +\frac {5}{2} a^{2} b^{8} B \right ) x^{18}+\left (\frac {1}{19} b^{10} A +\frac {10}{19} a \,b^{9} B \right ) x^{19}+\frac {b^{10} B \,x^{20}}{20}\) | \(235\) |
default | \(\frac {b^{10} B \,x^{20}}{20}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{19}}{19}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{18}}{18}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{17}}{17}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{16}}{16}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{15}}{15}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{14}}{14}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{13}}{13}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{12}}{12}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{11}}{11}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{10}}{10}+\frac {a^{10} A \,x^{9}}{9}\) | \(244\) |
gosper | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
risch | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
parallelrisch | \(\frac {1}{9} a^{10} A \,x^{9}+x^{10} a^{9} b A +\frac {1}{10} x^{10} a^{10} B +\frac {45}{11} x^{11} a^{8} b^{2} A +\frac {10}{11} x^{11} a^{9} b B +10 x^{12} a^{7} b^{3} A +\frac {15}{4} x^{12} a^{8} b^{2} B +\frac {210}{13} x^{13} a^{6} b^{4} A +\frac {120}{13} x^{13} a^{7} b^{3} B +18 A \,a^{5} b^{5} x^{14}+15 B \,a^{6} b^{4} x^{14}+14 x^{15} a^{4} b^{6} A +\frac {84}{5} x^{15} a^{5} b^{5} B +\frac {15}{2} x^{16} a^{3} b^{7} A +\frac {105}{8} x^{16} a^{4} b^{6} B +\frac {45}{17} x^{17} a^{2} b^{8} A +\frac {120}{17} x^{17} a^{3} b^{7} B +\frac {5}{9} x^{18} a \,b^{9} A +\frac {5}{2} x^{18} a^{2} b^{8} B +\frac {1}{19} x^{19} b^{10} A +\frac {10}{19} x^{19} a \,b^{9} B +\frac {1}{20} b^{10} B \,x^{20}\) | \(245\) |
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Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {1}{9} \, A a^{10} x^{9} + \frac {1}{19} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac {5}{18} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac {15}{17} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac {15}{8} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac {14}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac {30}{13} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac {5}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac {5}{11} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac {1}{10} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \]
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Time = 0.04 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.10 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{9}}{9} + \frac {B b^{10} x^{20}}{20} + x^{19} \left (\frac {A b^{10}}{19} + \frac {10 B a b^{9}}{19}\right ) + x^{18} \cdot \left (\frac {5 A a b^{9}}{9} + \frac {5 B a^{2} b^{8}}{2}\right ) + x^{17} \cdot \left (\frac {45 A a^{2} b^{8}}{17} + \frac {120 B a^{3} b^{7}}{17}\right ) + x^{16} \cdot \left (\frac {15 A a^{3} b^{7}}{2} + \frac {105 B a^{4} b^{6}}{8}\right ) + x^{15} \cdot \left (14 A a^{4} b^{6} + \frac {84 B a^{5} b^{5}}{5}\right ) + x^{14} \cdot \left (18 A a^{5} b^{5} + 15 B a^{6} b^{4}\right ) + x^{13} \cdot \left (\frac {210 A a^{6} b^{4}}{13} + \frac {120 B a^{7} b^{3}}{13}\right ) + x^{12} \cdot \left (10 A a^{7} b^{3} + \frac {15 B a^{8} b^{2}}{4}\right ) + x^{11} \cdot \left (\frac {45 A a^{8} b^{2}}{11} + \frac {10 B a^{9} b}{11}\right ) + x^{10} \left (A a^{9} b + \frac {B a^{10}}{10}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {1}{9} \, A a^{10} x^{9} + \frac {1}{19} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac {5}{18} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac {15}{17} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac {15}{8} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac {14}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac {30}{13} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac {5}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac {5}{11} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac {1}{10} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \]
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Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=\frac {1}{20} \, B b^{10} x^{20} + \frac {10}{19} \, B a b^{9} x^{19} + \frac {1}{19} \, A b^{10} x^{19} + \frac {5}{2} \, B a^{2} b^{8} x^{18} + \frac {5}{9} \, A a b^{9} x^{18} + \frac {120}{17} \, B a^{3} b^{7} x^{17} + \frac {45}{17} \, A a^{2} b^{8} x^{17} + \frac {105}{8} \, B a^{4} b^{6} x^{16} + \frac {15}{2} \, A a^{3} b^{7} x^{16} + \frac {84}{5} \, B a^{5} b^{5} x^{15} + 14 \, A a^{4} b^{6} x^{15} + 15 \, B a^{6} b^{4} x^{14} + 18 \, A a^{5} b^{5} x^{14} + \frac {120}{13} \, B a^{7} b^{3} x^{13} + \frac {210}{13} \, A a^{6} b^{4} x^{13} + \frac {15}{4} \, B a^{8} b^{2} x^{12} + 10 \, A a^{7} b^{3} x^{12} + \frac {10}{11} \, B a^{9} b x^{11} + \frac {45}{11} \, A a^{8} b^{2} x^{11} + \frac {1}{10} \, B a^{10} x^{10} + A a^{9} b x^{10} + \frac {1}{9} \, A a^{10} x^{9} \]
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Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.88 \[ \int x^8 (a+b x)^{10} (A+B x) \, dx=x^{10}\,\left (\frac {B\,a^{10}}{10}+A\,b\,a^9\right )+x^{19}\,\left (\frac {A\,b^{10}}{19}+\frac {10\,B\,a\,b^9}{19}\right )+\frac {A\,a^{10}\,x^9}{9}+\frac {B\,b^{10}\,x^{20}}{20}+\frac {5\,a^7\,b^2\,x^{12}\,\left (8\,A\,b+3\,B\,a\right )}{4}+\frac {30\,a^6\,b^3\,x^{13}\,\left (7\,A\,b+4\,B\,a\right )}{13}+3\,a^5\,b^4\,x^{14}\,\left (6\,A\,b+5\,B\,a\right )+\frac {14\,a^4\,b^5\,x^{15}\,\left (5\,A\,b+6\,B\,a\right )}{5}+\frac {15\,a^3\,b^6\,x^{16}\,\left (4\,A\,b+7\,B\,a\right )}{8}+\frac {15\,a^2\,b^7\,x^{17}\,\left (3\,A\,b+8\,B\,a\right )}{17}+\frac {5\,a^8\,b\,x^{11}\,\left (9\,A\,b+2\,B\,a\right )}{11}+\frac {5\,a\,b^8\,x^{18}\,\left (2\,A\,b+9\,B\,a\right )}{18} \]
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