Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {a x^{1+m} \sqrt {d+e x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{d (1+m)}+b \text {Int}\left (\frac {x^m \arctan (c x)}{\sqrt {d+e x^2}},x\right ) \]
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Not integrable
Time = 0.10 (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 = {} \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = a \int \frac {x^m}{\sqrt {d+e x^2}} \, dx+b \int \frac {x^m \arctan (c x)}{\sqrt {d+e x^2}} \, dx \\ & = b \int \frac {x^m \arctan (c x)}{\sqrt {d+e x^2}} \, dx+\frac {\left (a \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {x^m}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{\sqrt {d+e x^2}} \\ & = \frac {a x^{1+m} \sqrt {1+\frac {e x^2}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right )}{(1+m) \sqrt {d+e x^2}}+b \int \frac {x^m \arctan (c x)}{\sqrt {d+e x^2}} \, dx \\ \end{align*}
Not integrable
Time = 3.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 22.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 0.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 0.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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