Integrand size = 16, antiderivative size = 216 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=-\frac {a^{10} A}{8 x^8}-\frac {a^9 (10 A b+a B)}{7 x^7}-\frac {5 a^8 b (9 A b+2 a B)}{6 x^6}-\frac {3 a^7 b^2 (8 A b+3 a B)}{x^5}-\frac {15 a^6 b^3 (7 A b+4 a B)}{2 x^4}-\frac {14 a^5 b^4 (6 A b+5 a B)}{x^3}-\frac {21 a^4 b^5 (5 A b+6 a B)}{x^2}-\frac {30 a^3 b^6 (4 A b+7 a B)}{x}+5 a b^8 (2 A b+9 a B) x+\frac {1}{2} b^9 (A b+10 a B) x^2+\frac {1}{3} b^{10} B x^3+15 a^2 b^7 (3 A b+8 a B) \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 216, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=-\frac {a^{10} A}{8 x^8}-\frac {a^9 (a B+10 A b)}{7 x^7}-\frac {5 a^8 b (2 a B+9 A b)}{6 x^6}-\frac {3 a^7 b^2 (3 a B+8 A b)}{x^5}-\frac {15 a^6 b^3 (4 a B+7 A b)}{2 x^4}-\frac {14 a^5 b^4 (5 a B+6 A b)}{x^3}-\frac {21 a^4 b^5 (6 a B+5 A b)}{x^2}-\frac {30 a^3 b^6 (7 a B+4 A b)}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)+\frac {1}{2} b^9 x^2 (10 a B+A b)+5 a b^8 x (9 a B+2 A b)+\frac {1}{3} b^{10} B x^3 \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (5 a b^8 (2 A b+9 a B)+\frac {a^{10} A}{x^9}+\frac {a^9 (10 A b+a B)}{x^8}+\frac {5 a^8 b (9 A b+2 a B)}{x^7}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^6}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^5}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^4}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^3}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x^2}+\frac {15 a^2 b^7 (3 A b+8 a B)}{x}+b^9 (A b+10 a B) x+b^{10} B x^2\right ) \, dx \\ & = -\frac {a^{10} A}{8 x^8}-\frac {a^9 (10 A b+a B)}{7 x^7}-\frac {5 a^8 b (9 A b+2 a B)}{6 x^6}-\frac {3 a^7 b^2 (8 A b+3 a B)}{x^5}-\frac {15 a^6 b^3 (7 A b+4 a B)}{2 x^4}-\frac {14 a^5 b^4 (6 A b+5 a B)}{x^3}-\frac {21 a^4 b^5 (5 A b+6 a B)}{x^2}-\frac {30 a^3 b^6 (4 A b+7 a B)}{x}+5 a b^8 (2 A b+9 a B) x+\frac {1}{2} b^9 (A b+10 a B) x^2+\frac {1}{3} b^{10} B x^3+15 a^2 b^7 (3 A b+8 a B) \log (x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=-\frac {120 a^3 A b^7}{x}+45 a^2 b^8 B x+5 a b^9 x (2 A+B x)-\frac {105 a^4 b^6 (A+2 B x)}{x^2}+\frac {1}{6} b^{10} x^2 (3 A+2 B x)-\frac {42 a^5 b^5 (2 A+3 B x)}{x^3}-\frac {35 a^6 b^4 (3 A+4 B x)}{2 x^4}-\frac {6 a^7 b^3 (4 A+5 B x)}{x^5}-\frac {3 a^8 b^2 (5 A+6 B x)}{2 x^6}-\frac {5 a^9 b (6 A+7 B x)}{21 x^7}-\frac {a^{10} (7 A+8 B x)}{56 x^8}+15 a^2 b^7 (3 A b+8 a B) \log (x) \]
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Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {b^{10} B \,x^{3}}{3}+\frac {A \,b^{10} x^{2}}{2}+5 B a \,b^{9} x^{2}+10 A a \,b^{9} x +45 B \,a^{2} b^{8} x +15 a^{2} b^{7} \left (3 A b +8 B a \right ) \ln \left (x \right )-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{6 x^{6}}-\frac {a^{9} \left (10 A b +B a \right )}{7 x^{7}}-\frac {a^{10} A}{8 x^{8}}-\frac {14 a^{5} b^{4} \left (6 A b +5 B a \right )}{x^{3}}-\frac {30 a^{3} b^{6} \left (4 A b +7 B a \right )}{x}-\frac {21 a^{4} b^{5} \left (5 A b +6 B a \right )}{x^{2}}-\frac {15 a^{6} b^{3} \left (7 A b +4 B a \right )}{2 x^{4}}-\frac {3 a^{7} b^{2} \left (8 A b +3 B a \right )}{x^{5}}\) | \(210\) |
risch | \(\frac {b^{10} B \,x^{3}}{3}+\frac {A \,b^{10} x^{2}}{2}+5 B a \,b^{9} x^{2}+10 A a \,b^{9} x +45 B \,a^{2} b^{8} x +\frac {\left (-120 a^{3} b^{7} A -210 a^{4} b^{6} B \right ) x^{7}+\left (-105 a^{4} b^{6} A -126 a^{5} b^{5} B \right ) x^{6}+\left (-84 a^{5} b^{5} A -70 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {105}{2} a^{6} b^{4} A -30 a^{7} b^{3} B \right ) x^{4}+\left (-24 a^{7} b^{3} A -9 a^{8} b^{2} B \right ) x^{3}+\left (-\frac {15}{2} a^{8} b^{2} A -\frac {5}{3} a^{9} b B \right ) x^{2}+\left (-\frac {10}{7} a^{9} b A -\frac {1}{7} a^{10} B \right ) x -\frac {a^{10} A}{8}}{x^{8}}+45 A \ln \left (x \right ) a^{2} b^{8}+120 B \ln \left (x \right ) a^{3} b^{7}\) | \(233\) |
norman | \(\frac {\left (\frac {1}{2} b^{10} A +5 a \,b^{9} B \right ) x^{10}+\left (-\frac {105}{2} a^{6} b^{4} A -30 a^{7} b^{3} B \right ) x^{4}+\left (-\frac {15}{2} a^{8} b^{2} A -\frac {5}{3} a^{9} b B \right ) x^{2}+\left (-\frac {10}{7} a^{9} b A -\frac {1}{7} a^{10} B \right ) x +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{9}+\left (-120 a^{3} b^{7} A -210 a^{4} b^{6} B \right ) x^{7}+\left (-105 a^{4} b^{6} A -126 a^{5} b^{5} B \right ) x^{6}+\left (-84 a^{5} b^{5} A -70 a^{6} b^{4} B \right ) x^{5}+\left (-24 a^{7} b^{3} A -9 a^{8} b^{2} B \right ) x^{3}-\frac {a^{10} A}{8}+\frac {b^{10} B \,x^{11}}{3}}{x^{8}}+\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) \ln \left (x \right )\) | \(235\) |
parallelrisch | \(\frac {56 b^{10} B \,x^{11}+84 A \,b^{10} x^{10}+840 B a \,b^{9} x^{10}+7560 A \ln \left (x \right ) x^{8} a^{2} b^{8}+1680 a A \,b^{9} x^{9}+20160 B \ln \left (x \right ) x^{8} a^{3} b^{7}+7560 B \,a^{2} b^{8} x^{9}-20160 a^{3} A \,b^{7} x^{7}-35280 B \,a^{4} b^{6} x^{7}-17640 a^{4} A \,b^{6} x^{6}-21168 B \,a^{5} b^{5} x^{6}-14112 a^{5} A \,b^{5} x^{5}-11760 B \,a^{6} b^{4} x^{5}-8820 a^{6} A \,b^{4} x^{4}-5040 B \,a^{7} b^{3} x^{4}-4032 a^{7} A \,b^{3} x^{3}-1512 B \,a^{8} b^{2} x^{3}-1260 a^{8} A \,b^{2} x^{2}-280 B \,a^{9} b \,x^{2}-240 a^{9} A b x -24 a^{10} B x -21 a^{10} A}{168 x^{8}}\) | \(248\) |
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Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=\frac {56 \, B b^{10} x^{11} - 21 \, A a^{10} + 84 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 840 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 2520 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} \log \left (x\right ) - 5040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 2352 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 1260 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 504 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 140 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 24 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]
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Time = 4.00 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=\frac {B b^{10} x^{3}}{3} + 15 a^{2} b^{7} \cdot \left (3 A b + 8 B a\right ) \log {\left (x \right )} + x^{2} \left (\frac {A b^{10}}{2} + 5 B a b^{9}\right ) + x \left (10 A a b^{9} + 45 B a^{2} b^{8}\right ) + \frac {- 21 A a^{10} + x^{7} \left (- 20160 A a^{3} b^{7} - 35280 B a^{4} b^{6}\right ) + x^{6} \left (- 17640 A a^{4} b^{6} - 21168 B a^{5} b^{5}\right ) + x^{5} \left (- 14112 A a^{5} b^{5} - 11760 B a^{6} b^{4}\right ) + x^{4} \left (- 8820 A a^{6} b^{4} - 5040 B a^{7} b^{3}\right ) + x^{3} \left (- 4032 A a^{7} b^{3} - 1512 B a^{8} b^{2}\right ) + x^{2} \left (- 1260 A a^{8} b^{2} - 280 B a^{9} b\right ) + x \left (- 240 A a^{9} b - 24 B a^{10}\right )}{168 x^{8}} \]
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Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=\frac {1}{3} \, B b^{10} x^{3} + \frac {1}{2} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{2} + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x + 15 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} \log \left (x\right ) - \frac {21 \, A a^{10} + 5040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]
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Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=\frac {1}{3} \, B b^{10} x^{3} + 5 \, B a b^{9} x^{2} + \frac {1}{2} \, A b^{10} x^{2} + 45 \, B a^{2} b^{8} x + 10 \, A a b^{9} x + 15 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} \log \left ({\left | x \right |}\right ) - \frac {21 \, A a^{10} + 5040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]
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Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^9} \, dx=x^2\,\left (\frac {A\,b^{10}}{2}+5\,B\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{7}+\frac {10\,A\,b\,a^9}{7}\right )+\frac {A\,a^{10}}{8}+x^2\,\left (\frac {5\,B\,a^9\,b}{3}+\frac {15\,A\,a^8\,b^2}{2}\right )+x^3\,\left (9\,B\,a^8\,b^2+24\,A\,a^7\,b^3\right )+x^4\,\left (30\,B\,a^7\,b^3+\frac {105\,A\,a^6\,b^4}{2}\right )+x^5\,\left (70\,B\,a^6\,b^4+84\,A\,a^5\,b^5\right )+x^6\,\left (126\,B\,a^5\,b^5+105\,A\,a^4\,b^6\right )+x^7\,\left (210\,B\,a^4\,b^6+120\,A\,a^3\,b^7\right )}{x^8}+\ln \left (x\right )\,\left (120\,B\,a^3\,b^7+45\,A\,a^2\,b^8\right )+\frac {B\,b^{10}\,x^3}{3}+5\,a\,b^8\,x\,\left (2\,A\,b+9\,B\,a\right ) \]
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