Integrand size = 11, antiderivative size = 119 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=-\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x) \]
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Time = 0.04 (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} \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=-\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+210 a^4 b^6 \log (x)+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (120 a^3 b^7+\frac {a^{10}}{x^7}+\frac {10 a^9 b}{x^6}+\frac {45 a^8 b^2}{x^5}+\frac {120 a^7 b^3}{x^4}+\frac {210 a^6 b^4}{x^3}+\frac {252 a^5 b^5}{x^2}+\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+10 a b^9 x^2+b^{10} x^3\right ) \, dx \\ & = -\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=-\frac {a^{10}}{6 x^6}-\frac {2 a^9 b}{x^5}-\frac {45 a^8 b^2}{4 x^4}-\frac {40 a^7 b^3}{x^3}-\frac {105 a^6 b^4}{x^2}-\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+\frac {45}{2} a^2 b^8 x^2+\frac {10}{3} a b^9 x^3+\frac {b^{10} x^4}{4}+210 a^4 b^6 \log (x) \]
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Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{10}}{6 x^{6}}-\frac {2 a^{9} b}{x^{5}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {40 a^{7} b^{3}}{x^{3}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {252 a^{5} b^{5}}{x}+120 a^{3} b^{7} x +\frac {45 a^{2} b^{8} x^{2}}{2}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {b^{10} x^{4}}{4}+210 a^{4} b^{6} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{4}}{4}+\frac {10 a \,b^{9} x^{3}}{3}+\frac {45 a^{2} b^{8} x^{2}}{2}+120 a^{3} b^{7} x +\frac {-252 a^{5} b^{5} x^{5}-105 a^{6} b^{4} x^{4}-40 a^{7} b^{3} x^{3}-\frac {45}{4} a^{8} b^{2} x^{2}-2 a^{9} b x -\frac {1}{6} a^{10}}{x^{6}}+210 a^{4} b^{6} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{6} a^{10}+\frac {1}{4} b^{10} x^{10}+\frac {10}{3} a \,b^{9} x^{9}+\frac {45}{2} a^{2} b^{8} x^{8}+120 a^{3} b^{7} x^{7}-252 a^{5} b^{5} x^{5}-105 a^{6} b^{4} x^{4}-40 a^{7} b^{3} x^{3}-\frac {45}{4} a^{8} b^{2} x^{2}-2 a^{9} b x}{x^{6}}+210 a^{4} b^{6} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {3 b^{10} x^{10}+40 a \,b^{9} x^{9}+270 a^{2} b^{8} x^{8}+2520 a^{4} b^{6} \ln \left (x \right ) x^{6}+1440 a^{3} b^{7} x^{7}-3024 a^{5} b^{5} x^{5}-1260 a^{6} b^{4} x^{4}-480 a^{7} b^{3} x^{3}-135 a^{8} b^{2} x^{2}-24 a^{9} b x -2 a^{10}}{12 x^{6}}\) | \(115\) |
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Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=\frac {3 \, b^{10} x^{10} + 40 \, a b^{9} x^{9} + 270 \, a^{2} b^{8} x^{8} + 1440 \, a^{3} b^{7} x^{7} + 2520 \, a^{4} b^{6} x^{6} \log \left (x\right ) - 3024 \, a^{5} b^{5} x^{5} - 1260 \, a^{6} b^{4} x^{4} - 480 \, a^{7} b^{3} x^{3} - 135 \, a^{8} b^{2} x^{2} - 24 \, a^{9} b x - 2 \, a^{10}}{12 \, x^{6}} \]
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Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=210 a^{4} b^{6} \log {\left (x \right )} + 120 a^{3} b^{7} x + \frac {45 a^{2} b^{8} x^{2}}{2} + \frac {10 a b^{9} x^{3}}{3} + \frac {b^{10} x^{4}}{4} + \frac {- 2 a^{10} - 24 a^{9} b x - 135 a^{8} b^{2} x^{2} - 480 a^{7} b^{3} x^{3} - 1260 a^{6} b^{4} x^{4} - 3024 a^{5} b^{5} x^{5}}{12 x^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=\frac {1}{4} \, b^{10} x^{4} + \frac {10}{3} \, a b^{9} x^{3} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 120 \, a^{3} b^{7} x + 210 \, a^{4} b^{6} \log \left (x\right ) - \frac {3024 \, a^{5} b^{5} x^{5} + 1260 \, a^{6} b^{4} x^{4} + 480 \, a^{7} b^{3} x^{3} + 135 \, a^{8} b^{2} x^{2} + 24 \, a^{9} b x + 2 \, a^{10}}{12 \, x^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=\frac {1}{4} \, b^{10} x^{4} + \frac {10}{3} \, a b^{9} x^{3} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 120 \, a^{3} b^{7} x + 210 \, a^{4} b^{6} \log \left ({\left | x \right |}\right ) - \frac {3024 \, a^{5} b^{5} x^{5} + 1260 \, a^{6} b^{4} x^{4} + 480 \, a^{7} b^{3} x^{3} + 135 \, a^{8} b^{2} x^{2} + 24 \, a^{9} b x + 2 \, a^{10}}{12 \, x^{6}} \]
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Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^7} \, dx=\frac {b^{10}\,x^4}{4}-\frac {\frac {a^{10}}{6}+2\,a^9\,b\,x+\frac {45\,a^8\,b^2\,x^2}{4}+40\,a^7\,b^3\,x^3+105\,a^6\,b^4\,x^4+252\,a^5\,b^5\,x^5}{x^6}+120\,a^3\,b^7\,x+\frac {10\,a\,b^9\,x^3}{3}+\frac {45\,a^2\,b^8\,x^2}{2}+210\,a^4\,b^6\,\ln \left (x\right ) \]
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