Integrand size = 18, antiderivative size = 18 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\frac {2 (d x)^{3/2} (a+b \arccos (c x))^3}{3 d}+\frac {2 b c \text {Int}\left (\frac {(d x)^{3/2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{d} \]
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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 = {} \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int \sqrt {d x} (a+b \arccos (c x))^3 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{3/2} (a+b \arccos (c x))^3}{3 d}+\frac {(2 b c) \int \frac {(d x)^{3/2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{d} \\ \end{align*}
Not integrable
Time = 141.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int \sqrt {d x} (a+b \arccos (c x))^3 \, dx \]
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Not integrable
Time = 1.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
\[\int \left (a +b \arccos \left (c x \right )\right )^{3} \sqrt {d x}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int { \sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{3} \,d x } \]
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Not integrable
Time = 8.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int \sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}\, dx \]
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Not integrable
Time = 3.42 (sec) , antiderivative size = 418, normalized size of antiderivative = 23.22 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int { \sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{3} \,d x } \]
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Exception generated. \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \sqrt {d x} (a+b \arccos (c x))^3 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3\,\sqrt {d\,x} \,d x \]
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