Integrand size = 32, antiderivative size = 222 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4763, 4741, 4737, 4723, 327, 222} \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {b^2 \arcsin (c x) \sqrt {c d x+d} \sqrt {e-c e x}}{4 c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 x \sqrt {c d x+d} \sqrt {e-c e x} \]
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Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4741
Rule 4763
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.30 \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-12 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (b \cos (2 \arcsin (c x))+2 a \sin (2 \arcsin (c x)))+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (2 a+b \sin (2 \arcsin (c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (4 a^2 c x \sqrt {1-c^2 x^2}+2 a b \cos (2 \arcsin (c x))-b^2 \sin (2 \arcsin (c x))\right )}{24 c \sqrt {1-c^2 x^2}} \]
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\[\int \sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
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\[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {c d x + d} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
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Exception generated. \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {c d x + d} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x} \,d x \]
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