Integrand size = 23, antiderivative size = 23 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))^4}{d e (1+m)}-\frac {4 b \text {Int}\left (\frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}},x\right )}{e (1+m)} \]
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Not integrable
Time = 0.13 (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 = {} \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^m (a+b \arcsin (x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))^4}{d e (1+m)}-\frac {(4 b) \text {Subst}\left (\int \frac {(e x)^{1+m} (a+b \arcsin (x))^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e (1+m)} \\ \end{align*}
Not integrable
Time = 1.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx \]
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Not integrable
Time = 2.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \left (d e x +c e \right )^{m} \left (a +b \arcsin \left (d x +c \right )\right )^{4}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} {\left (d e x + c e\right )}^{m} \,d x } \]
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Not integrable
Time = 52.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}\, dx \]
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Not integrable
Time = 10.10 (sec) , antiderivative size = 618, normalized size of antiderivative = 26.87 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} {\left (d e x + c e\right )}^{m} \,d x } \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} {\left (d e x + c e\right )}^{m} \,d x } \]
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Not integrable
Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \]
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