Integrand size = 11, antiderivative size = 117 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (210 a^4 b^6+\frac {a^{10}}{x^6}+\frac {10 a^9 b}{x^5}+\frac {45 a^8 b^2}{x^4}+\frac {120 a^7 b^3}{x^3}+\frac {210 a^6 b^4}{x^2}+\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+45 a^2 b^8 x^2+10 a b^9 x^3+b^{10} x^4\right ) \, dx \\ & = -\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \]
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Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {a^{10}}{5 x^{5}}-\frac {5 a^{9} b}{2 x^{4}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {60 a^{7} b^{3}}{x^{2}}-\frac {210 a^{6} b^{4}}{x}+210 a^{4} b^{6} x +60 a^{3} b^{7} x^{2}+15 a^{2} b^{8} x^{3}+\frac {5 a \,b^{9} x^{4}}{2}+\frac {b^{10} x^{5}}{5}+252 a^{5} b^{5} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{5}}{5}+\frac {5 a \,b^{9} x^{4}}{2}+15 a^{2} b^{8} x^{3}+60 a^{3} b^{7} x^{2}+210 a^{4} b^{6} x +\frac {-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x -\frac {1}{5} a^{10}}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{5} a^{10}+\frac {1}{5} b^{10} x^{10}+\frac {5}{2} a \,b^{9} x^{9}+15 a^{2} b^{8} x^{8}+60 a^{3} b^{7} x^{7}+210 a^{4} b^{6} x^{6}-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {2 b^{10} x^{10}+25 a \,b^{9} x^{9}+150 a^{2} b^{8} x^{8}+600 a^{3} b^{7} x^{7}+2520 a^{5} b^{5} \ln \left (x \right ) x^{5}+2100 a^{4} b^{6} x^{6}-2100 a^{6} b^{4} x^{4}-600 a^{7} b^{3} x^{3}-150 a^{8} b^{2} x^{2}-25 a^{9} b x -2 a^{10}}{10 x^{5}}\) | \(115\) |
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Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {2 \, b^{10} x^{10} + 25 \, a b^{9} x^{9} + 150 \, a^{2} b^{8} x^{8} + 600 \, a^{3} b^{7} x^{7} + 2100 \, a^{4} b^{6} x^{6} + 2520 \, a^{5} b^{5} x^{5} \log \left (x\right ) - 2100 \, a^{6} b^{4} x^{4} - 600 \, a^{7} b^{3} x^{3} - 150 \, a^{8} b^{2} x^{2} - 25 \, a^{9} b x - 2 \, a^{10}}{10 \, x^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=252 a^{5} b^{5} \log {\left (x \right )} + 210 a^{4} b^{6} x + 60 a^{3} b^{7} x^{2} + 15 a^{2} b^{8} x^{3} + \frac {5 a b^{9} x^{4}}{2} + \frac {b^{10} x^{5}}{5} + \frac {- 2 a^{10} - 25 a^{9} b x - 150 a^{8} b^{2} x^{2} - 600 a^{7} b^{3} x^{3} - 2100 a^{6} b^{4} x^{4}}{10 x^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left (x\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left ({\left | x \right |}\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \]
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Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {b^{10}\,x^5}{5}-\frac {\frac {a^{10}}{5}+\frac {5\,a^9\,b\,x}{2}+15\,a^8\,b^2\,x^2+60\,a^7\,b^3\,x^3+210\,a^6\,b^4\,x^4}{x^5}+210\,a^4\,b^6\,x+\frac {5\,a\,b^9\,x^4}{2}+60\,a^3\,b^7\,x^2+15\,a^2\,b^8\,x^3+252\,a^5\,b^5\,\ln \left (x\right ) \]
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