Integrand size = 13, antiderivative size = 27 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+2 a b \log (x)+\frac {b^2 x^7}{7} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \text {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,x^7\right ) \\ & = \frac {1}{7} \text {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^7\right ) \\ & = -\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \]
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Time = 5.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {a^{2}}{7 x^{7}}+\frac {b^{2} x^{7}}{7}+2 a b \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {a^{2}}{7 x^{7}}+\frac {b^{2} x^{7}}{7}+2 a b \ln \left (x \right )\) | \(24\) |
norman | \(\frac {-\frac {a^{2}}{7}+\frac {b^{2} x^{14}}{7}}{x^{7}}+2 a b \ln \left (x \right )\) | \(26\) |
parallelrisch | \(\frac {b^{2} x^{14}+14 a b \ln \left (x \right ) x^{7}-a^{2}}{7 x^{7}}\) | \(28\) |
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none
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {b^{2} x^{14} + 14 \, a b x^{7} \log \left (x\right ) - a^{2}}{7 \, x^{7}} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=- \frac {a^{2}}{7 x^{7}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{7}}{7} \]
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none
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a b \log \left (x^{7}\right ) - \frac {a^{2}}{7 \, x^{7}} \]
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none
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {1}{7} \, b^{2} x^{7} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {2 \, a b x^{7} + a^{2}}{7 \, x^{7}} \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {b^2\,x^7}{7}-\frac {a^2}{7\,x^7}+2\,a\,b\,\ln \left (x\right ) \]
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