Integrand size = 18, antiderivative size = 18 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\frac {2 (d x)^{5/2} (a+b \arccos (c x))^3}{5 d}+\frac {6 b c \text {Int}\left (\frac {(d x)^{5/2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{5 d} \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 18, 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 = {} \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{5/2} (a+b \arccos (c x))^3}{5 d}+\frac {(6 b c) \int \frac {(d x)^{5/2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{5 d} \\ \end{align*}
Not integrable
Time = 45.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx \]
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Not integrable
Time = 0.82 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
\[\int \left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{3}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.94 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{3} \,d x } \]
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Not integrable
Time = 80.91 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}\, dx \]
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Not integrable
Time = 3.50 (sec) , antiderivative size = 441, normalized size of antiderivative = 24.50 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{3} \,d x } \]
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Exception generated. \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (d x)^{3/2} (a+b \arccos (c x))^3 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3\,{\left (d\,x\right )}^{3/2} \,d x \]
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