Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a d e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \text {Int}\left (\frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4},x\right ) \]
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Not integrable
Time = 0.13 (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 = {} \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = a \int \frac {\left (d+e x^2\right )^{5/2}}{x^4} \, dx+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx \\ & = -\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx+\frac {1}{3} (5 a e) \int \frac {\left (d+e x^2\right )^{3/2}}{x^2} \, dx \\ & = -\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx+\left (5 a e^2\right ) \int \sqrt {d+e x^2} \, dx \\ & = \frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx+\frac {1}{2} \left (5 a d e^2\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx \\ & = \frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx+\frac {1}{2} \left (5 a d e^2\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right ) \\ & = \frac {5}{2} a e^2 x \sqrt {d+e x^2}-\frac {5 a e \left (d+e x^2\right )^{3/2}}{3 x}-\frac {a \left (d+e x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a d e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \int \frac {\left (d+e x^2\right )^{5/2} \arctan (c x)}{x^4} \, dx \\ \end{align*}
Not integrable
Time = 11.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx \]
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Not integrable
Time = 0.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{4}}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 42.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}{x^{4}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2}}{x^4} \,d x \]
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