Integrand size = 20, antiderivative size = 20 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\text {Int}\left (\frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx \\ \end{align*}
Not integrable
Time = 51.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx \]
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Not integrable
Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {{\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}}{e x +d}d x\]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{e x + d} \,d x } \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{e x + d} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{d+e\,x} \,d x \]
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