Integrand size = 18, antiderivative size = 18 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=-\frac {2 (a+b \arccos (c x))^3}{d \sqrt {d x}}-\frac {6 b c \text {Int}\left (\frac {(a+b \arccos (c x))^2}{\sqrt {d x} \sqrt {1-c^2 x^2}},x\right )}{d} \]
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Not integrable
Time = 0.09 (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 = {} \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b \arccos (c x))^3}{d \sqrt {d x}}-\frac {(6 b c) \int \frac {(a+b \arccos (c x))^2}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d} \\ \end{align*}
Not integrable
Time = 51.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx \]
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Not integrable
Time = 2.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{\left (d x \right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 3.41 (sec) , antiderivative size = 489, normalized size of antiderivative = 27.17 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.80 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^3}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{{\left (d\,x\right )}^{3/2}} \,d x \]
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