Integrand size = 11, antiderivative size = 30 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]
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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} \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{x^8}+\frac {2 a b}{x^7}+\frac {b^2}{x^6}\right ) \, dx \\ & = -\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]
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Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80
method | result | size |
norman | \(\frac {-\frac {1}{5} b^{2} x^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\) | \(24\) |
risch | \(\frac {-\frac {1}{5} b^{2} x^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\) | \(24\) |
gosper | \(-\frac {21 b^{2} x^{2}+35 a b x +15 a^{2}}{105 x^{7}}\) | \(25\) |
default | \(-\frac {a^{2}}{7 x^{7}}-\frac {a b}{3 x^{6}}-\frac {b^{2}}{5 x^{5}}\) | \(25\) |
parallelrisch | \(\frac {-21 b^{2} x^{2}-35 a b x -15 a^{2}}{105 x^{7}}\) | \(25\) |
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^2}{x^8} \, dx=\frac {- 15 a^{2} - 35 a b x - 21 b^{2} x^{2}}{105 x^{7}} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^2}{x^8} \, dx=-\frac {\frac {a^2}{7}+\frac {a\,b\,x}{3}+\frac {b^2\,x^2}{5}}{x^7} \]
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