Integrand size = 11, antiderivative size = 115 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=-\frac {a^{10}}{7 x^7}-\frac {5 a^9 b}{3 x^6}-\frac {9 a^8 b^2}{x^5}-\frac {30 a^7 b^3}{x^4}-\frac {70 a^6 b^4}{x^3}-\frac {126 a^5 b^5}{x^2}-\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+5 a b^9 x^2+\frac {b^{10} x^3}{3}+120 a^3 b^7 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 115, 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} \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=-\frac {a^{10}}{7 x^7}-\frac {5 a^9 b}{3 x^6}-\frac {9 a^8 b^2}{x^5}-\frac {30 a^7 b^3}{x^4}-\frac {70 a^6 b^4}{x^3}-\frac {126 a^5 b^5}{x^2}-\frac {210 a^4 b^6}{x}+120 a^3 b^7 \log (x)+45 a^2 b^8 x+5 a b^9 x^2+\frac {b^{10} x^3}{3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (45 a^2 b^8+\frac {a^{10}}{x^8}+\frac {10 a^9 b}{x^7}+\frac {45 a^8 b^2}{x^6}+\frac {120 a^7 b^3}{x^5}+\frac {210 a^6 b^4}{x^4}+\frac {252 a^5 b^5}{x^3}+\frac {210 a^4 b^6}{x^2}+\frac {120 a^3 b^7}{x}+10 a b^9 x+b^{10} x^2\right ) \, dx \\ & = -\frac {a^{10}}{7 x^7}-\frac {5 a^9 b}{3 x^6}-\frac {9 a^8 b^2}{x^5}-\frac {30 a^7 b^3}{x^4}-\frac {70 a^6 b^4}{x^3}-\frac {126 a^5 b^5}{x^2}-\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+5 a b^9 x^2+\frac {b^{10} x^3}{3}+120 a^3 b^7 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=-\frac {a^{10}}{7 x^7}-\frac {5 a^9 b}{3 x^6}-\frac {9 a^8 b^2}{x^5}-\frac {30 a^7 b^3}{x^4}-\frac {70 a^6 b^4}{x^3}-\frac {126 a^5 b^5}{x^2}-\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+5 a b^9 x^2+\frac {b^{10} x^3}{3}+120 a^3 b^7 \log (x) \]
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Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {a^{10}}{7 x^{7}}-\frac {5 a^{9} b}{3 x^{6}}-\frac {9 a^{8} b^{2}}{x^{5}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {126 a^{5} b^{5}}{x^{2}}-\frac {210 a^{4} b^{6}}{x}+45 a^{2} b^{8} x +5 a \,b^{9} x^{2}+\frac {b^{10} x^{3}}{3}+120 a^{3} b^{7} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{3}}{3}+5 a \,b^{9} x^{2}+45 a^{2} b^{8} x +\frac {-210 a^{4} b^{6} x^{6}-126 a^{5} b^{5} x^{5}-70 a^{6} b^{4} x^{4}-30 a^{7} b^{3} x^{3}-9 a^{8} b^{2} x^{2}-\frac {5}{3} a^{9} b x -\frac {1}{7} a^{10}}{x^{7}}+120 a^{3} b^{7} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{7} a^{10}+\frac {1}{3} b^{10} x^{10}+5 a \,b^{9} x^{9}+45 a^{2} b^{8} x^{8}-210 a^{4} b^{6} x^{6}-126 a^{5} b^{5} x^{5}-70 a^{6} b^{4} x^{4}-30 a^{7} b^{3} x^{3}-9 a^{8} b^{2} x^{2}-\frac {5}{3} a^{9} b x}{x^{7}}+120 a^{3} b^{7} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {7 b^{10} x^{10}+105 a \,b^{9} x^{9}+2520 a^{3} b^{7} \ln \left (x \right ) x^{7}+945 a^{2} b^{8} x^{8}-4410 a^{4} b^{6} x^{6}-2646 a^{5} b^{5} x^{5}-1470 a^{6} b^{4} x^{4}-630 a^{7} b^{3} x^{3}-189 a^{8} b^{2} x^{2}-35 a^{9} b x -3 a^{10}}{21 x^{7}}\) | \(115\) |
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Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=\frac {7 \, b^{10} x^{10} + 105 \, a b^{9} x^{9} + 945 \, a^{2} b^{8} x^{8} + 2520 \, a^{3} b^{7} x^{7} \log \left (x\right ) - 4410 \, a^{4} b^{6} x^{6} - 2646 \, a^{5} b^{5} x^{5} - 1470 \, a^{6} b^{4} x^{4} - 630 \, a^{7} b^{3} x^{3} - 189 \, a^{8} b^{2} x^{2} - 35 \, a^{9} b x - 3 \, a^{10}}{21 \, x^{7}} \]
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=120 a^{3} b^{7} \log {\left (x \right )} + 45 a^{2} b^{8} x + 5 a b^{9} x^{2} + \frac {b^{10} x^{3}}{3} + \frac {- 3 a^{10} - 35 a^{9} b x - 189 a^{8} b^{2} x^{2} - 630 a^{7} b^{3} x^{3} - 1470 a^{6} b^{4} x^{4} - 2646 a^{5} b^{5} x^{5} - 4410 a^{4} b^{6} x^{6}}{21 x^{7}} \]
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Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=\frac {1}{3} \, b^{10} x^{3} + 5 \, a b^{9} x^{2} + 45 \, a^{2} b^{8} x + 120 \, a^{3} b^{7} \log \left (x\right ) - \frac {4410 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 1470 \, a^{6} b^{4} x^{4} + 630 \, a^{7} b^{3} x^{3} + 189 \, a^{8} b^{2} x^{2} + 35 \, a^{9} b x + 3 \, a^{10}}{21 \, x^{7}} \]
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Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=\frac {1}{3} \, b^{10} x^{3} + 5 \, a b^{9} x^{2} + 45 \, a^{2} b^{8} x + 120 \, a^{3} b^{7} \log \left ({\left | x \right |}\right ) - \frac {4410 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 1470 \, a^{6} b^{4} x^{4} + 630 \, a^{7} b^{3} x^{3} + 189 \, a^{8} b^{2} x^{2} + 35 \, a^{9} b x + 3 \, a^{10}}{21 \, x^{7}} \]
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Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^8} \, dx=\frac {b^{10}\,x^3}{3}-\frac {\frac {a^{10}}{7}+\frac {5\,a^9\,b\,x}{3}+9\,a^8\,b^2\,x^2+30\,a^7\,b^3\,x^3+70\,a^6\,b^4\,x^4+126\,a^5\,b^5\,x^5+210\,a^4\,b^6\,x^6}{x^7}+45\,a^2\,b^8\,x+5\,a\,b^9\,x^2+120\,a^3\,b^7\,\ln \left (x\right ) \]
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