Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))^3}{d e}-\frac {6 b \text {Int}\left (\frac {\sqrt {e (c+d x)} (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}},x\right )}{e} \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arcsin (x))^3}{\sqrt {e x}} \, dx,x,c+d x\right )}{d} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))^3}{d e}-\frac {(6 b) \text {Subst}\left (\int \frac {\sqrt {e x} (a+b \arcsin (x))^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e} \\ \end{align*}
Not integrable
Time = 91.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{3}}{\sqrt {d e x +c e}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{\sqrt {d e x + c e}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{\sqrt {d e x + c e}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{\sqrt {c\,e+d\,e\,x}} \,d x \]
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