Integrand size = 21, antiderivative size = 21 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=-\frac {6 c^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{140 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{28 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}+\frac {3}{35} c^3 \text {Int}\left (\frac {1}{\sqrt {\arctan (a x)}},x\right )+\frac {9}{280} c^2 \text {Int}\left (\frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}},x\right )+\frac {1}{56} c \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}},x\right )+\frac {16}{35} c^3 \text {Int}\left (\arctan (a x)^{3/2},x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 21, 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 = {} \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{28 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2} \, dx \\ & = -\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{140 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{28 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{280} \left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx \\ & = -\frac {6 c^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{140 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{28 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{280} \left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{35} \left (3 c^3\right ) \int \frac {1}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{35} \left (16 c^3\right ) \int \arctan (a x)^{3/2} \, dx \\ \end{align*}
Not integrable
Time = 1.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx \]
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Not integrable
Time = 3.60 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
\[\int \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 18.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.14 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 76.85 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{\frac {3}{2}} \,d x } \]
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Not integrable
Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
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