Integrand size = 16, antiderivative size = 229 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {a^{10} A}{18 x^{18}}-\frac {a^9 (10 A b+a B)}{17 x^{17}}-\frac {5 a^8 b (9 A b+2 a B)}{16 x^{16}}-\frac {a^7 b^2 (8 A b+3 a B)}{x^{15}}-\frac {15 a^6 b^3 (7 A b+4 a B)}{7 x^{14}}-\frac {42 a^5 b^4 (6 A b+5 a B)}{13 x^{13}}-\frac {7 a^4 b^5 (5 A b+6 a B)}{2 x^{12}}-\frac {30 a^3 b^6 (4 A b+7 a B)}{11 x^{11}}-\frac {3 a^2 b^7 (3 A b+8 a B)}{2 x^{10}}-\frac {5 a b^8 (2 A b+9 a B)}{9 x^9}-\frac {b^9 (A b+10 a B)}{8 x^8}-\frac {b^{10} B}{7 x^7} \]
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Time = 0.09 (sec) , antiderivative size = 229, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {a^{10} A}{18 x^{18}}-\frac {a^9 (a B+10 A b)}{17 x^{17}}-\frac {5 a^8 b (2 a B+9 A b)}{16 x^{16}}-\frac {a^7 b^2 (3 a B+8 A b)}{x^{15}}-\frac {15 a^6 b^3 (4 a B+7 A b)}{7 x^{14}}-\frac {42 a^5 b^4 (5 a B+6 A b)}{13 x^{13}}-\frac {7 a^4 b^5 (6 a B+5 A b)}{2 x^{12}}-\frac {30 a^3 b^6 (7 a B+4 A b)}{11 x^{11}}-\frac {3 a^2 b^7 (8 a B+3 A b)}{2 x^{10}}-\frac {b^9 (10 a B+A b)}{8 x^8}-\frac {5 a b^8 (9 a B+2 A b)}{9 x^9}-\frac {b^{10} B}{7 x^7} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^{10} A}{x^{19}}+\frac {a^9 (10 A b+a B)}{x^{18}}+\frac {5 a^8 b (9 A b+2 a B)}{x^{17}}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^{16}}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^{15}}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^{14}}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^{13}}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x^{12}}+\frac {15 a^2 b^7 (3 A b+8 a B)}{x^{11}}+\frac {5 a b^8 (2 A b+9 a B)}{x^{10}}+\frac {b^9 (A b+10 a B)}{x^9}+\frac {b^{10} B}{x^8}\right ) \, dx \\ & = -\frac {a^{10} A}{18 x^{18}}-\frac {a^9 (10 A b+a B)}{17 x^{17}}-\frac {5 a^8 b (9 A b+2 a B)}{16 x^{16}}-\frac {a^7 b^2 (8 A b+3 a B)}{x^{15}}-\frac {15 a^6 b^3 (7 A b+4 a B)}{7 x^{14}}-\frac {42 a^5 b^4 (6 A b+5 a B)}{13 x^{13}}-\frac {7 a^4 b^5 (5 A b+6 a B)}{2 x^{12}}-\frac {30 a^3 b^6 (4 A b+7 a B)}{11 x^{11}}-\frac {3 a^2 b^7 (3 A b+8 a B)}{2 x^{10}}-\frac {5 a b^8 (2 A b+9 a B)}{9 x^9}-\frac {b^9 (A b+10 a B)}{8 x^8}-\frac {b^{10} B}{7 x^7} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {b^{10} (7 A+8 B x)}{56 x^8}-\frac {5 a b^9 (8 A+9 B x)}{36 x^9}-\frac {a^2 b^8 (9 A+10 B x)}{2 x^{10}}-\frac {12 a^3 b^7 (10 A+11 B x)}{11 x^{11}}-\frac {35 a^4 b^6 (11 A+12 B x)}{22 x^{12}}-\frac {21 a^5 b^5 (12 A+13 B x)}{13 x^{13}}-\frac {15 a^6 b^4 (13 A+14 B x)}{13 x^{14}}-\frac {4 a^7 b^3 (14 A+15 B x)}{7 x^{15}}-\frac {3 a^8 b^2 (15 A+16 B x)}{16 x^{16}}-\frac {5 a^9 b (16 A+17 B x)}{136 x^{17}}-\frac {a^{10} (17 A+18 B x)}{306 x^{18}} \]
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Time = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {a^{10} A}{18 x^{18}}-\frac {a^{9} \left (10 A b +B a \right )}{17 x^{17}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{16 x^{16}}-\frac {a^{7} b^{2} \left (8 A b +3 B a \right )}{x^{15}}-\frac {15 a^{6} b^{3} \left (7 A b +4 B a \right )}{7 x^{14}}-\frac {42 a^{5} b^{4} \left (6 A b +5 B a \right )}{13 x^{13}}-\frac {7 a^{4} b^{5} \left (5 A b +6 B a \right )}{2 x^{12}}-\frac {30 a^{3} b^{6} \left (4 A b +7 B a \right )}{11 x^{11}}-\frac {3 a^{2} b^{7} \left (3 A b +8 B a \right )}{2 x^{10}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{9 x^{9}}-\frac {b^{9} \left (A b +10 B a \right )}{8 x^{8}}-\frac {b^{10} B}{7 x^{7}}\) | \(208\) |
norman | \(\frac {-\frac {a^{10} A}{18}+\left (-\frac {10}{17} a^{9} b A -\frac {1}{17} a^{10} B \right ) x +\left (-\frac {45}{16} a^{8} b^{2} A -\frac {5}{8} a^{9} b B \right ) x^{2}+\left (-8 a^{7} b^{3} A -3 a^{8} b^{2} B \right ) x^{3}+\left (-15 a^{6} b^{4} A -\frac {60}{7} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {252}{13} a^{5} b^{5} A -\frac {210}{13} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {35}{2} a^{4} b^{6} A -21 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{11} a^{3} b^{7} A -\frac {210}{11} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {9}{2} a^{2} b^{8} A -12 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {10}{9} a \,b^{9} A -5 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{8} b^{10} A -\frac {5}{4} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{7}}{x^{18}}\) | \(235\) |
risch | \(\frac {-\frac {a^{10} A}{18}+\left (-\frac {10}{17} a^{9} b A -\frac {1}{17} a^{10} B \right ) x +\left (-\frac {45}{16} a^{8} b^{2} A -\frac {5}{8} a^{9} b B \right ) x^{2}+\left (-8 a^{7} b^{3} A -3 a^{8} b^{2} B \right ) x^{3}+\left (-15 a^{6} b^{4} A -\frac {60}{7} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {252}{13} a^{5} b^{5} A -\frac {210}{13} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {35}{2} a^{4} b^{6} A -21 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{11} a^{3} b^{7} A -\frac {210}{11} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {9}{2} a^{2} b^{8} A -12 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {10}{9} a \,b^{9} A -5 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{8} b^{10} A -\frac {5}{4} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{7}}{x^{18}}\) | \(235\) |
gosper | \(-\frac {350064 b^{10} B \,x^{11}+306306 A \,b^{10} x^{10}+3063060 B a \,b^{9} x^{10}+2722720 a A \,b^{9} x^{9}+12252240 B \,a^{2} b^{8} x^{9}+11027016 a^{2} A \,b^{8} x^{8}+29405376 B \,a^{3} b^{7} x^{8}+26732160 a^{3} A \,b^{7} x^{7}+46781280 B \,a^{4} b^{6} x^{7}+42882840 a^{4} A \,b^{6} x^{6}+51459408 B \,a^{5} b^{5} x^{6}+47500992 a^{5} A \,b^{5} x^{5}+39584160 B \,a^{6} b^{4} x^{5}+36756720 a^{6} A \,b^{4} x^{4}+21003840 B \,a^{7} b^{3} x^{4}+19603584 a^{7} A \,b^{3} x^{3}+7351344 B \,a^{8} b^{2} x^{3}+6891885 a^{8} A \,b^{2} x^{2}+1531530 B \,a^{9} b \,x^{2}+1441440 a^{9} A b x +144144 a^{10} B x +136136 a^{10} A}{2450448 x^{18}}\) | \(244\) |
parallelrisch | \(-\frac {350064 b^{10} B \,x^{11}+306306 A \,b^{10} x^{10}+3063060 B a \,b^{9} x^{10}+2722720 a A \,b^{9} x^{9}+12252240 B \,a^{2} b^{8} x^{9}+11027016 a^{2} A \,b^{8} x^{8}+29405376 B \,a^{3} b^{7} x^{8}+26732160 a^{3} A \,b^{7} x^{7}+46781280 B \,a^{4} b^{6} x^{7}+42882840 a^{4} A \,b^{6} x^{6}+51459408 B \,a^{5} b^{5} x^{6}+47500992 a^{5} A \,b^{5} x^{5}+39584160 B \,a^{6} b^{4} x^{5}+36756720 a^{6} A \,b^{4} x^{4}+21003840 B \,a^{7} b^{3} x^{4}+19603584 a^{7} A \,b^{3} x^{3}+7351344 B \,a^{8} b^{2} x^{3}+6891885 a^{8} A \,b^{2} x^{2}+1531530 B \,a^{9} b \,x^{2}+1441440 a^{9} A b x +144144 a^{10} B x +136136 a^{10} A}{2450448 x^{18}}\) | \(244\) |
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Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]
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Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]
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Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {350064 \, B b^{10} x^{11} + 3063060 \, B a b^{9} x^{10} + 306306 \, A b^{10} x^{10} + 12252240 \, B a^{2} b^{8} x^{9} + 2722720 \, A a b^{9} x^{9} + 29405376 \, B a^{3} b^{7} x^{8} + 11027016 \, A a^{2} b^{8} x^{8} + 46781280 \, B a^{4} b^{6} x^{7} + 26732160 \, A a^{3} b^{7} x^{7} + 51459408 \, B a^{5} b^{5} x^{6} + 42882840 \, A a^{4} b^{6} x^{6} + 39584160 \, B a^{6} b^{4} x^{5} + 47500992 \, A a^{5} b^{5} x^{5} + 21003840 \, B a^{7} b^{3} x^{4} + 36756720 \, A a^{6} b^{4} x^{4} + 7351344 \, B a^{8} b^{2} x^{3} + 19603584 \, A a^{7} b^{3} x^{3} + 1531530 \, B a^{9} b x^{2} + 6891885 \, A a^{8} b^{2} x^{2} + 144144 \, B a^{10} x + 1441440 \, A a^{9} b x + 136136 \, A a^{10}}{2450448 \, x^{18}} \]
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Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{19}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{17}+\frac {10\,A\,b\,a^9}{17}\right )+\frac {A\,a^{10}}{18}+x^9\,\left (5\,B\,a^2\,b^8+\frac {10\,A\,a\,b^9}{9}\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{8}+\frac {45\,A\,a^8\,b^2}{16}\right )+x^{10}\,\left (\frac {A\,b^{10}}{8}+\frac {5\,B\,a\,b^9}{4}\right )+x^3\,\left (3\,B\,a^8\,b^2+8\,A\,a^7\,b^3\right )+x^8\,\left (12\,B\,a^3\,b^7+\frac {9\,A\,a^2\,b^8}{2}\right )+x^6\,\left (21\,B\,a^5\,b^5+\frac {35\,A\,a^4\,b^6}{2}\right )+x^4\,\left (\frac {60\,B\,a^7\,b^3}{7}+15\,A\,a^6\,b^4\right )+x^7\,\left (\frac {210\,B\,a^4\,b^6}{11}+\frac {120\,A\,a^3\,b^7}{11}\right )+x^5\,\left (\frac {210\,B\,a^6\,b^4}{13}+\frac {252\,A\,a^5\,b^5}{13}\right )+\frac {B\,b^{10}\,x^{11}}{7}}{x^{18}} \]
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