Integrand size = 16, antiderivative size = 148 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=10 a^9 A b x+\frac {45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac {105}{2} a^6 A b^4 x^4+\frac {252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac {120}{7} a^3 A b^7 x^7+\frac {45}{8} a^2 A b^8 x^8+\frac {10}{9} a A b^9 x^9+\frac {1}{10} A b^{10} x^{10}+\frac {B (a+b x)^{11}}{11 b}+a^{10} A \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 148, 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 = {81, 45} \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=a^{10} A \log (x)+10 a^9 A b x+\frac {45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac {105}{2} a^6 A b^4 x^4+\frac {252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac {120}{7} a^3 A b^7 x^7+\frac {45}{8} a^2 A b^8 x^8+\frac {10}{9} a A b^9 x^9+\frac {B (a+b x)^{11}}{11 b}+\frac {1}{10} A b^{10} x^{10} \]
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Rule 45
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b x)^{11}}{11 b}+A \int \frac {(a+b x)^{10}}{x} \, dx \\ & = \frac {B (a+b x)^{11}}{11 b}+A \int \left (10 a^9 b+\frac {a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx \\ & = 10 a^9 A b x+\frac {45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac {105}{2} a^6 A b^4 x^4+\frac {252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac {120}{7} a^3 A b^7 x^7+\frac {45}{8} a^2 A b^8 x^8+\frac {10}{9} a A b^9 x^9+\frac {1}{10} A b^{10} x^{10}+\frac {B (a+b x)^{11}}{11 b}+a^{10} A \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=a^{10} B x+5 a^9 b x (2 A+B x)+\frac {15}{2} a^8 b^2 x^2 (3 A+2 B x)+10 a^7 b^3 x^3 (4 A+3 B x)+\frac {21}{2} a^6 b^4 x^4 (5 A+4 B x)+\frac {42}{5} a^5 b^5 x^5 (6 A+5 B x)+5 a^4 b^6 x^6 (7 A+6 B x)+\frac {15}{7} a^3 b^7 x^7 (8 A+7 B x)+\frac {5}{8} a^2 b^8 x^8 (9 A+8 B x)+\frac {1}{9} a b^9 x^9 (10 A+9 B x)+\frac {1}{110} b^{10} x^{10} (11 A+10 B x)+a^{10} A \log (x) \]
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Time = 0.40 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.55
method | result | size |
norman | \(\left (\frac {1}{10} b^{10} A +a \,b^{9} B \right ) x^{10}+\left (\frac {10}{9} a \,b^{9} A +5 a^{2} b^{8} B \right ) x^{9}+\left (\frac {45}{8} a^{2} b^{8} A +15 a^{3} b^{7} B \right ) x^{8}+\left (\frac {120}{7} a^{3} b^{7} A +30 a^{4} b^{6} B \right ) x^{7}+\left (\frac {252}{5} a^{5} b^{5} A +42 a^{6} b^{4} B \right ) x^{5}+\left (\frac {105}{2} a^{6} b^{4} A +30 a^{7} b^{3} B \right ) x^{4}+\left (\frac {45}{2} a^{8} b^{2} A +5 a^{9} b B \right ) x^{2}+\left (35 a^{4} b^{6} A +42 a^{5} b^{5} B \right ) x^{6}+\left (40 a^{7} b^{3} A +15 a^{8} b^{2} B \right ) x^{3}+\left (10 a^{9} b A +a^{10} B \right ) x +\frac {b^{10} B \,x^{11}}{11}+a^{10} A \ln \left (x \right )\) | \(230\) |
default | \(\frac {b^{10} B \,x^{11}}{11}+\frac {A \,b^{10} x^{10}}{10}+B a \,b^{9} x^{10}+\frac {10 a A \,b^{9} x^{9}}{9}+5 B \,a^{2} b^{8} x^{9}+\frac {45 a^{2} A \,b^{8} x^{8}}{8}+15 B \,a^{3} b^{7} x^{8}+\frac {120 a^{3} A \,b^{7} x^{7}}{7}+30 B \,a^{4} b^{6} x^{7}+35 a^{4} A \,b^{6} x^{6}+42 B \,a^{5} b^{5} x^{6}+\frac {252 a^{5} A \,b^{5} x^{5}}{5}+42 B \,a^{6} b^{4} x^{5}+\frac {105 a^{6} A \,b^{4} x^{4}}{2}+30 B \,a^{7} b^{3} x^{4}+40 a^{7} A \,b^{3} x^{3}+15 B \,a^{8} b^{2} x^{3}+\frac {45 a^{8} A \,b^{2} x^{2}}{2}+5 B \,a^{9} b \,x^{2}+10 a^{9} A b x +a^{10} B x +a^{10} A \ln \left (x \right )\) | \(238\) |
risch | \(\frac {b^{10} B \,x^{11}}{11}+\frac {A \,b^{10} x^{10}}{10}+B a \,b^{9} x^{10}+\frac {10 a A \,b^{9} x^{9}}{9}+5 B \,a^{2} b^{8} x^{9}+\frac {45 a^{2} A \,b^{8} x^{8}}{8}+15 B \,a^{3} b^{7} x^{8}+\frac {120 a^{3} A \,b^{7} x^{7}}{7}+30 B \,a^{4} b^{6} x^{7}+35 a^{4} A \,b^{6} x^{6}+42 B \,a^{5} b^{5} x^{6}+\frac {252 a^{5} A \,b^{5} x^{5}}{5}+42 B \,a^{6} b^{4} x^{5}+\frac {105 a^{6} A \,b^{4} x^{4}}{2}+30 B \,a^{7} b^{3} x^{4}+40 a^{7} A \,b^{3} x^{3}+15 B \,a^{8} b^{2} x^{3}+\frac {45 a^{8} A \,b^{2} x^{2}}{2}+5 B \,a^{9} b \,x^{2}+10 a^{9} A b x +a^{10} B x +a^{10} A \ln \left (x \right )\) | \(238\) |
parallelrisch | \(\frac {b^{10} B \,x^{11}}{11}+\frac {A \,b^{10} x^{10}}{10}+B a \,b^{9} x^{10}+\frac {10 a A \,b^{9} x^{9}}{9}+5 B \,a^{2} b^{8} x^{9}+\frac {45 a^{2} A \,b^{8} x^{8}}{8}+15 B \,a^{3} b^{7} x^{8}+\frac {120 a^{3} A \,b^{7} x^{7}}{7}+30 B \,a^{4} b^{6} x^{7}+35 a^{4} A \,b^{6} x^{6}+42 B \,a^{5} b^{5} x^{6}+\frac {252 a^{5} A \,b^{5} x^{5}}{5}+42 B \,a^{6} b^{4} x^{5}+\frac {105 a^{6} A \,b^{4} x^{4}}{2}+30 B \,a^{7} b^{3} x^{4}+40 a^{7} A \,b^{3} x^{3}+15 B \,a^{8} b^{2} x^{3}+\frac {45 a^{8} A \,b^{2} x^{2}}{2}+5 B \,a^{9} b \,x^{2}+10 a^{9} A b x +a^{10} B x +a^{10} A \ln \left (x \right )\) | \(238\) |
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Time = 0.22 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=\frac {1}{11} \, B b^{10} x^{11} + A a^{10} \log \left (x\right ) + \frac {1}{10} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac {5}{9} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac {15}{8} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac {30}{7} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac {42}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac {15}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + {\left (B a^{10} + 10 \, A a^{9} b\right )} x \]
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Time = 0.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=A a^{10} \log {\left (x \right )} + \frac {B b^{10} x^{11}}{11} + x^{10} \left (\frac {A b^{10}}{10} + B a b^{9}\right ) + x^{9} \cdot \left (\frac {10 A a b^{9}}{9} + 5 B a^{2} b^{8}\right ) + x^{8} \cdot \left (\frac {45 A a^{2} b^{8}}{8} + 15 B a^{3} b^{7}\right ) + x^{7} \cdot \left (\frac {120 A a^{3} b^{7}}{7} + 30 B a^{4} b^{6}\right ) + x^{6} \cdot \left (35 A a^{4} b^{6} + 42 B a^{5} b^{5}\right ) + x^{5} \cdot \left (\frac {252 A a^{5} b^{5}}{5} + 42 B a^{6} b^{4}\right ) + x^{4} \cdot \left (\frac {105 A a^{6} b^{4}}{2} + 30 B a^{7} b^{3}\right ) + x^{3} \cdot \left (40 A a^{7} b^{3} + 15 B a^{8} b^{2}\right ) + x^{2} \cdot \left (\frac {45 A a^{8} b^{2}}{2} + 5 B a^{9} b\right ) + x \left (10 A a^{9} b + B a^{10}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=\frac {1}{11} \, B b^{10} x^{11} + A a^{10} \log \left (x\right ) + \frac {1}{10} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac {5}{9} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac {15}{8} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac {30}{7} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac {42}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac {15}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + {\left (B a^{10} + 10 \, A a^{9} b\right )} x \]
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Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=\frac {1}{11} \, B b^{10} x^{11} + B a b^{9} x^{10} + \frac {1}{10} \, A b^{10} x^{10} + 5 \, B a^{2} b^{8} x^{9} + \frac {10}{9} \, A a b^{9} x^{9} + 15 \, B a^{3} b^{7} x^{8} + \frac {45}{8} \, A a^{2} b^{8} x^{8} + 30 \, B a^{4} b^{6} x^{7} + \frac {120}{7} \, A a^{3} b^{7} x^{7} + 42 \, B a^{5} b^{5} x^{6} + 35 \, A a^{4} b^{6} x^{6} + 42 \, B a^{6} b^{4} x^{5} + \frac {252}{5} \, A a^{5} b^{5} x^{5} + 30 \, B a^{7} b^{3} x^{4} + \frac {105}{2} \, A a^{6} b^{4} x^{4} + 15 \, B a^{8} b^{2} x^{3} + 40 \, A a^{7} b^{3} x^{3} + 5 \, B a^{9} b x^{2} + \frac {45}{2} \, A a^{8} b^{2} x^{2} + B a^{10} x + 10 \, A a^{9} b x + A a^{10} \log \left ({\left | x \right |}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx=x\,\left (B\,a^{10}+10\,A\,b\,a^9\right )+x^{10}\,\left (\frac {A\,b^{10}}{10}+B\,a\,b^9\right )+\frac {B\,b^{10}\,x^{11}}{11}+A\,a^{10}\,\ln \left (x\right )+5\,a^7\,b^2\,x^3\,\left (8\,A\,b+3\,B\,a\right )+\frac {15\,a^6\,b^3\,x^4\,\left (7\,A\,b+4\,B\,a\right )}{2}+\frac {42\,a^5\,b^4\,x^5\,\left (6\,A\,b+5\,B\,a\right )}{5}+7\,a^4\,b^5\,x^6\,\left (5\,A\,b+6\,B\,a\right )+\frac {30\,a^3\,b^6\,x^7\,\left (4\,A\,b+7\,B\,a\right )}{7}+\frac {15\,a^2\,b^7\,x^8\,\left (3\,A\,b+8\,B\,a\right )}{8}+\frac {5\,a^8\,b\,x^2\,\left (9\,A\,b+2\,B\,a\right )}{2}+\frac {5\,a\,b^8\,x^9\,\left (2\,A\,b+9\,B\,a\right )}{9} \]
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