Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=a d^2 \sqrt {d+e x^2}+\frac {1}{3} a d \left (d+e x^2\right )^{3/2}+\frac {1}{5} a \left (d+e x^2\right )^{5/2}-a d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \text {Int}\left (\frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x},x\right ) \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = a \int \frac {\left (d+e x^2\right )^{5/2}}{x} \, dx+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {(d+e x)^{5/2}}{x} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx \\ & = \frac {1}{5} a \left (d+e x^2\right )^{5/2}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx+\frac {1}{2} (a d) \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{3} a d \left (d+e x^2\right )^{3/2}+\frac {1}{5} a \left (d+e x^2\right )^{5/2}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx+\frac {1}{2} \left (a d^2\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right ) \\ & = a d^2 \sqrt {d+e x^2}+\frac {1}{3} a d \left (d+e x^2\right )^{3/2}+\frac {1}{5} a \left (d+e x^2\right )^{5/2}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx+\frac {1}{2} \left (a d^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right ) \\ & = a d^2 \sqrt {d+e x^2}+\frac {1}{3} a d \left (d+e x^2\right )^{3/2}+\frac {1}{5} a \left (d+e x^2\right )^{5/2}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx+\frac {\left (a d^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e} \\ & = a d^2 \sqrt {d+e x^2}+\frac {1}{3} a d \left (d+e x^2\right )^{3/2}+\frac {1}{5} a \left (d+e x^2\right )^{5/2}-a d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 10.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )}{x}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )}}{x} \,d x } \]
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Not integrable
Time = 40.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}{x}\, dx \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2}}{x} \,d x \]
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