Integrand size = 11, antiderivative size = 30 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \]
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Time = 0.01 (sec) , antiderivative size = 30, 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} \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{x^{10}}+\frac {2 a b}{x^9}+\frac {b^2}{x^8}\right ) \, dx \\ & = -\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80
method | result | size |
norman | \(\frac {-\frac {1}{7} b^{2} x^{2}-\frac {1}{4} a b x -\frac {1}{9} a^{2}}{x^{9}}\) | \(24\) |
risch | \(\frac {-\frac {1}{7} b^{2} x^{2}-\frac {1}{4} a b x -\frac {1}{9} a^{2}}{x^{9}}\) | \(24\) |
gosper | \(-\frac {36 b^{2} x^{2}+63 a b x +28 a^{2}}{252 x^{9}}\) | \(25\) |
default | \(-\frac {a^{2}}{9 x^{9}}-\frac {a b}{4 x^{8}}-\frac {b^{2}}{7 x^{7}}\) | \(25\) |
parallelrisch | \(\frac {-36 b^{2} x^{2}-63 a b x -28 a^{2}}{252 x^{9}}\) | \(25\) |
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=\frac {- 28 a^{2} - 63 a b x - 36 b^{2} x^{2}}{252 x^{9}} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^{10}} \, dx=-\frac {\frac {a^2}{9}+\frac {a\,b\,x}{4}+\frac {b^2\,x^2}{7}}{x^9} \]
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