Integrand size = 40, antiderivative size = 14 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15 \log \left (x^2\right )}{\log (3+2 x)} \]
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\[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{x (3+2 x) \log ^2(3+2 x)} \, dx \\ & = \int \left (-\frac {30 \log \left (x^2\right )}{(3+2 x) \log ^2(3+2 x)}+\frac {30}{x \log (3+2 x)}\right ) \, dx \\ & = -\left (30 \int \frac {\log \left (x^2\right )}{(3+2 x) \log ^2(3+2 x)} \, dx\right )+30 \int \frac {1}{x \log (3+2 x)} \, dx \\ & = -\left (15 \text {Subst}\left (\int \frac {\log \left (\left (-\frac {3}{2}+\frac {x}{2}\right )^2\right )}{x \log ^2(x)} \, dx,x,3+2 x\right )\right )+30 \int \frac {1}{x \log (3+2 x)} \, dx \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15 \log \left (x^2\right )}{\log (3+2 x)} \]
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Time = 1.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {15 \ln \left (x^{2}\right )}{\ln \left (3+2 x \right )}\) | \(15\) |
risch | \(\frac {30 \ln \left (x \right )-\frac {15 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+15 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\frac {15 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}}{\ln \left (3+2 x \right )}\) | \(65\) |
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15 \, \log \left (x^{2}\right )}{\log \left (2 \, x + 3\right )} \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15 \log {\left (x^{2} \right )}}{\log {\left (2 x + 3 \right )}} \]
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Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {30 \, \log \left (x\right )}{\log \left (2 \, x + 3\right )} \]
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Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15 \, \log \left (x^{2}\right )}{\log \left (2 \, x + 3\right )} \]
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Time = 12.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-30 x \log \left (x^2\right )+(90+60 x) \log (3+2 x)}{\left (3 x+2 x^2\right ) \log ^2(3+2 x)} \, dx=\frac {15\,\ln \left (x^2\right )}{\ln \left (2\,x+3\right )} \]
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