Integrand size = 16, antiderivative size = 153 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=-\frac {a^{10} B}{10 x^{10}}-\frac {10 a^9 b B}{9 x^9}-\frac {45 a^8 b^2 B}{8 x^8}-\frac {120 a^7 b^3 B}{7 x^7}-\frac {35 a^6 b^4 B}{x^6}-\frac {252 a^5 b^5 B}{5 x^5}-\frac {105 a^4 b^6 B}{2 x^4}-\frac {40 a^3 b^7 B}{x^3}-\frac {45 a^2 b^8 B}{2 x^2}-\frac {10 a b^9 B}{x}-\frac {A (a+b x)^{11}}{11 a x^{11}}+b^{10} B \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 45} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=-\frac {a^{10} B}{10 x^{10}}-\frac {10 a^9 b B}{9 x^9}-\frac {45 a^8 b^2 B}{8 x^8}-\frac {120 a^7 b^3 B}{7 x^7}-\frac {35 a^6 b^4 B}{x^6}-\frac {252 a^5 b^5 B}{5 x^5}-\frac {105 a^4 b^6 B}{2 x^4}-\frac {40 a^3 b^7 B}{x^3}-\frac {45 a^2 b^8 B}{2 x^2}-\frac {A (a+b x)^{11}}{11 a x^{11}}-\frac {10 a b^9 B}{x}+b^{10} B \log (x) \]
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Rule 45
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{11}}{11 a x^{11}}+B \int \frac {(a+b x)^{10}}{x^{11}} \, dx \\ & = -\frac {A (a+b x)^{11}}{11 a x^{11}}+B \int \left (\frac {a^{10}}{x^{11}}+\frac {10 a^9 b}{x^{10}}+\frac {45 a^8 b^2}{x^9}+\frac {120 a^7 b^3}{x^8}+\frac {210 a^6 b^4}{x^7}+\frac {252 a^5 b^5}{x^6}+\frac {210 a^4 b^6}{x^5}+\frac {120 a^3 b^7}{x^4}+\frac {45 a^2 b^8}{x^3}+\frac {10 a b^9}{x^2}+\frac {b^{10}}{x}\right ) \, dx \\ & = -\frac {a^{10} B}{10 x^{10}}-\frac {10 a^9 b B}{9 x^9}-\frac {45 a^8 b^2 B}{8 x^8}-\frac {120 a^7 b^3 B}{7 x^7}-\frac {35 a^6 b^4 B}{x^6}-\frac {252 a^5 b^5 B}{5 x^5}-\frac {105 a^4 b^6 B}{2 x^4}-\frac {40 a^3 b^7 B}{x^3}-\frac {45 a^2 b^8 B}{2 x^2}-\frac {10 a b^9 B}{x}-\frac {A (a+b x)^{11}}{11 a x^{11}}+b^{10} B \log (x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=-\frac {A b^{10}}{x}-\frac {5 a b^9 (A+2 B x)}{x^2}-\frac {15 a^2 b^8 (2 A+3 B x)}{2 x^3}-\frac {10 a^3 b^7 (3 A+4 B x)}{x^4}-\frac {21 a^4 b^6 (4 A+5 B x)}{2 x^5}-\frac {42 a^5 b^5 (5 A+6 B x)}{5 x^6}-\frac {5 a^6 b^4 (6 A+7 B x)}{x^7}-\frac {15 a^7 b^3 (7 A+8 B x)}{7 x^8}-\frac {5 a^8 b^2 (8 A+9 B x)}{8 x^9}-\frac {a^9 b (9 A+10 B x)}{9 x^{10}}-\frac {a^{10} (10 A+11 B x)}{110 x^{11}}+b^{10} B \log (x) \]
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Time = 0.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.35
method | result | size |
default | \(b^{10} B \ln \left (x \right )-\frac {7 a^{5} b^{4} \left (6 A b +5 B a \right )}{x^{6}}-\frac {30 a^{6} b^{3} \left (7 A b +4 B a \right )}{7 x^{7}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{8 x^{8}}-\frac {a^{9} \left (10 A b +B a \right )}{10 x^{10}}-\frac {5 a^{2} b^{7} \left (3 A b +8 B a \right )}{x^{3}}-\frac {b^{9} \left (A b +10 B a \right )}{x}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{2 x^{2}}-\frac {15 a^{3} b^{6} \left (4 A b +7 B a \right )}{2 x^{4}}-\frac {42 a^{4} b^{5} \left (5 A b +6 B a \right )}{5 x^{5}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{9 x^{9}}-\frac {a^{10} A}{11 x^{11}}\) | \(206\) |
norman | \(\frac {\left (-5 a \,b^{9} A -\frac {45}{2} a^{2} b^{8} B \right ) x^{9}+\left (-30 a^{3} b^{7} A -\frac {105}{2} a^{4} b^{6} B \right ) x^{7}+\left (-42 a^{4} b^{6} A -\frac {252}{5} a^{5} b^{5} B \right ) x^{6}+\left (-30 a^{6} b^{4} A -\frac {120}{7} a^{7} b^{3} B \right ) x^{4}+\left (-15 a^{7} b^{3} A -\frac {45}{8} a^{8} b^{2} B \right ) x^{3}+\left (-5 a^{8} b^{2} A -\frac {10}{9} a^{9} b B \right ) x^{2}+\left (-a^{9} b A -\frac {1}{10} a^{10} B \right ) x +\left (-b^{10} A -10 a \,b^{9} B \right ) x^{10}+\left (-15 a^{2} b^{8} A -40 a^{3} b^{7} B \right ) x^{8}+\left (-42 a^{5} b^{5} A -35 a^{6} b^{4} B \right ) x^{5}-\frac {a^{10} A}{11}}{x^{11}}+b^{10} B \ln \left (x \right )\) | \(234\) |
risch | \(\frac {\left (-5 a \,b^{9} A -\frac {45}{2} a^{2} b^{8} B \right ) x^{9}+\left (-30 a^{3} b^{7} A -\frac {105}{2} a^{4} b^{6} B \right ) x^{7}+\left (-42 a^{4} b^{6} A -\frac {252}{5} a^{5} b^{5} B \right ) x^{6}+\left (-30 a^{6} b^{4} A -\frac {120}{7} a^{7} b^{3} B \right ) x^{4}+\left (-15 a^{7} b^{3} A -\frac {45}{8} a^{8} b^{2} B \right ) x^{3}+\left (-5 a^{8} b^{2} A -\frac {10}{9} a^{9} b B \right ) x^{2}+\left (-a^{9} b A -\frac {1}{10} a^{10} B \right ) x +\left (-b^{10} A -10 a \,b^{9} B \right ) x^{10}+\left (-15 a^{2} b^{8} A -40 a^{3} b^{7} B \right ) x^{8}+\left (-42 a^{5} b^{5} A -35 a^{6} b^{4} B \right ) x^{5}-\frac {a^{10} A}{11}}{x^{11}}+b^{10} B \ln \left (x \right )\) | \(234\) |
parallelrisch | \(-\frac {-27720 b^{10} B \ln \left (x \right ) x^{11}+27720 A \,b^{10} x^{10}+277200 B a \,b^{9} x^{10}+138600 a A \,b^{9} x^{9}+623700 B \,a^{2} b^{8} x^{9}+415800 a^{2} A \,b^{8} x^{8}+1108800 B \,a^{3} b^{7} x^{8}+831600 a^{3} A \,b^{7} x^{7}+1455300 B \,a^{4} b^{6} x^{7}+1164240 a^{4} A \,b^{6} x^{6}+1397088 B \,a^{5} b^{5} x^{6}+1164240 a^{5} A \,b^{5} x^{5}+970200 B \,a^{6} b^{4} x^{5}+831600 a^{6} A \,b^{4} x^{4}+475200 B \,a^{7} b^{3} x^{4}+415800 a^{7} A \,b^{3} x^{3}+155925 B \,a^{8} b^{2} x^{3}+138600 a^{8} A \,b^{2} x^{2}+30800 B \,a^{9} b \,x^{2}+27720 a^{9} A b x +2772 a^{10} B x +2520 a^{10} A}{27720 x^{11}}\) | \(246\) |
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Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=\frac {27720 \, B b^{10} x^{11} \log \left (x\right ) - 2520 \, A a^{10} - 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} - 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]
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Time = 26.80 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log {\left (x \right )} + \frac {- 2520 A a^{10} + x^{10} \left (- 27720 A b^{10} - 277200 B a b^{9}\right ) + x^{9} \left (- 138600 A a b^{9} - 623700 B a^{2} b^{8}\right ) + x^{8} \left (- 415800 A a^{2} b^{8} - 1108800 B a^{3} b^{7}\right ) + x^{7} \left (- 831600 A a^{3} b^{7} - 1455300 B a^{4} b^{6}\right ) + x^{6} \left (- 1164240 A a^{4} b^{6} - 1397088 B a^{5} b^{5}\right ) + x^{5} \left (- 1164240 A a^{5} b^{5} - 970200 B a^{6} b^{4}\right ) + x^{4} \left (- 831600 A a^{6} b^{4} - 475200 B a^{7} b^{3}\right ) + x^{3} \left (- 415800 A a^{7} b^{3} - 155925 B a^{8} b^{2}\right ) + x^{2} \left (- 138600 A a^{8} b^{2} - 30800 B a^{9} b\right ) + x \left (- 27720 A a^{9} b - 2772 B a^{10}\right )}{27720 x^{11}} \]
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Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log \left (x\right ) - \frac {2520 \, A a^{10} + 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]
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Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log \left ({\left | x \right |}\right ) - \frac {2520 \, A a^{10} + 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]
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Time = 0.41 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B\,b^{10}\,\ln \left (x\right )-\frac {x\,\left (\frac {B\,a^{10}}{10}+A\,b\,a^9\right )+\frac {A\,a^{10}}{11}+x^2\,\left (\frac {10\,B\,a^9\,b}{9}+5\,A\,a^8\,b^2\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{2}+5\,A\,a\,b^9\right )+x^{10}\,\left (A\,b^{10}+10\,B\,a\,b^9\right )+x^8\,\left (40\,B\,a^3\,b^7+15\,A\,a^2\,b^8\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{8}+15\,A\,a^7\,b^3\right )+x^5\,\left (35\,B\,a^6\,b^4+42\,A\,a^5\,b^5\right )+x^7\,\left (\frac {105\,B\,a^4\,b^6}{2}+30\,A\,a^3\,b^7\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{7}+30\,A\,a^6\,b^4\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{5}+42\,A\,a^4\,b^6\right )}{x^{11}} \]
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