Integrand size = 11, antiderivative size = 119 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6}+210 a^6 b^4 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 119, 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} \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (252 a^5 b^5+\frac {a^{10}}{x^5}+\frac {10 a^9 b}{x^4}+\frac {45 a^8 b^2}{x^3}+\frac {120 a^7 b^3}{x^2}+\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+120 a^3 b^7 x^2+45 a^2 b^8 x^3+10 a b^9 x^4+b^{10} x^5\right ) \, dx \\ & = -\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6}+210 a^6 b^4 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6}+210 a^6 b^4 \log (x) \]
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Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{10}}{4 x^{4}}-\frac {10 a^{9} b}{3 x^{3}}-\frac {45 a^{8} b^{2}}{2 x^{2}}-\frac {120 a^{7} b^{3}}{x}+252 a^{5} b^{5} x +105 a^{4} b^{6} x^{2}+40 a^{3} b^{7} x^{3}+\frac {45 a^{2} b^{8} x^{4}}{4}+2 a \,b^{9} x^{5}+\frac {b^{10} x^{6}}{6}+210 a^{6} b^{4} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{6}}{6}+2 a \,b^{9} x^{5}+\frac {45 a^{2} b^{8} x^{4}}{4}+40 a^{3} b^{7} x^{3}+105 a^{4} b^{6} x^{2}+252 a^{5} b^{5} x +\frac {-120 a^{7} b^{3} x^{3}-\frac {45}{2} a^{8} b^{2} x^{2}-\frac {10}{3} a^{9} b x -\frac {1}{4} a^{10}}{x^{4}}+210 a^{6} b^{4} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{4} a^{10}+\frac {1}{6} b^{10} x^{10}+2 a \,b^{9} x^{9}+\frac {45}{4} a^{2} b^{8} x^{8}+40 a^{3} b^{7} x^{7}+105 a^{4} b^{6} x^{6}+252 a^{5} b^{5} x^{5}-120 a^{7} b^{3} x^{3}-\frac {45}{2} a^{8} b^{2} x^{2}-\frac {10}{3} a^{9} b x}{x^{4}}+210 a^{6} b^{4} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {2 b^{10} x^{10}+24 a \,b^{9} x^{9}+135 a^{2} b^{8} x^{8}+480 a^{3} b^{7} x^{7}+1260 a^{4} b^{6} x^{6}+2520 a^{6} b^{4} \ln \left (x \right ) x^{4}+3024 a^{5} b^{5} x^{5}-1440 a^{7} b^{3} x^{3}-270 a^{8} b^{2} x^{2}-40 a^{9} b x -3 a^{10}}{12 x^{4}}\) | \(115\) |
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Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=\frac {2 \, b^{10} x^{10} + 24 \, a b^{9} x^{9} + 135 \, a^{2} b^{8} x^{8} + 480 \, a^{3} b^{7} x^{7} + 1260 \, a^{4} b^{6} x^{6} + 3024 \, a^{5} b^{5} x^{5} + 2520 \, a^{6} b^{4} x^{4} \log \left (x\right ) - 1440 \, a^{7} b^{3} x^{3} - 270 \, a^{8} b^{2} x^{2} - 40 \, a^{9} b x - 3 \, a^{10}}{12 \, x^{4}} \]
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Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=210 a^{6} b^{4} \log {\left (x \right )} + 252 a^{5} b^{5} x + 105 a^{4} b^{6} x^{2} + 40 a^{3} b^{7} x^{3} + \frac {45 a^{2} b^{8} x^{4}}{4} + 2 a b^{9} x^{5} + \frac {b^{10} x^{6}}{6} + \frac {- 3 a^{10} - 40 a^{9} b x - 270 a^{8} b^{2} x^{2} - 1440 a^{7} b^{3} x^{3}}{12 x^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=\frac {1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4} \log \left (x\right ) - \frac {1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=\frac {1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4} \log \left ({\left | x \right |}\right ) - \frac {1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^5} \, dx=\frac {b^{10}\,x^6}{6}-\frac {\frac {a^{10}}{4}+\frac {10\,a^9\,b\,x}{3}+\frac {45\,a^8\,b^2\,x^2}{2}+120\,a^7\,b^3\,x^3}{x^4}+252\,a^5\,b^5\,x+2\,a\,b^9\,x^5+105\,a^4\,b^6\,x^2+40\,a^3\,b^7\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+210\,a^6\,b^4\,\ln \left (x\right ) \]
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