Integrand size = 78, antiderivative size = 24 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {46}{3}+\frac {x^2}{\frac {x}{625}+\log \left (x+25 x^2\right )} \]
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\[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {625 x \left (-625-31249 x+25 x^2+1250 (1+25 x) \log (x (1+25 x))\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx \\ & = 625 \int \frac {x \left (-625-31249 x+25 x^2+1250 (1+25 x) \log (x (1+25 x))\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx \\ & = 625 \int \left (-\frac {x \left (625+31251 x+25 x^2\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2}+\frac {2 x}{x+625 \log (x (1+25 x))}\right ) \, dx \\ & = -\left (625 \int \frac {x \left (625+31251 x+25 x^2\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx \\ & = -\left (625 \int \left (-\frac {25}{(x+625 \log (x (1+25 x)))^2}+\frac {1250 x}{(x+625 \log (x (1+25 x)))^2}+\frac {x^2}{(x+625 \log (x (1+25 x)))^2}+\frac {25}{(1+25 x) (x+625 \log (x (1+25 x)))^2}\right ) \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx \\ & = -\left (625 \int \frac {x^2}{(x+625 \log (x (1+25 x)))^2} \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx+15625 \int \frac {1}{(x+625 \log (x (1+25 x)))^2} \, dx-15625 \int \frac {1}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx-781250 \int \frac {x}{(x+625 \log (x (1+25 x)))^2} \, dx \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 x^2}{x+625 \log (x (1+25 x))} \]
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Time = 2.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
norman | \(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\) | \(20\) |
risch | \(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\) | \(20\) |
parallelrisch | \(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\) | \(20\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x^{2} + x\right )} \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {x^{2}}{\frac {x}{625} + \log {\left (25 x^{2} + x \right )}} \]
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x + 1\right ) + 625 \, \log \left (x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x^{2} + x\right )} \]
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Time = 9.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625\,\left (2500\,x+1562500\,\ln \left (25\,x^2+x\right )+x^2\right )}{x+625\,\ln \left (25\,x^2+x\right )} \]
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