Integrand size = 32, antiderivative size = 231 \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {2 a b d x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 d \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 d x \sqrt {1-c^2 x^2} \arcsin (c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4763, 4847, 4737, 4767, 4715, 267} \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 a b d x \sqrt {1-c^2 x^2}}{\sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 d x \sqrt {1-c^2 x^2} \arcsin (c x)}{\sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 d \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rule 267
Rule 4715
Rule 4737
Rule 4763
Rule 4767
Rule 4847
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(d+c d x) (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (\frac {d (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {c d x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {\left (d \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (c d \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = -\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b d \sqrt {1-c^2 x^2}\right ) \int (a+b \arcsin (c x)) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {2 a b d x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 d \sqrt {1-c^2 x^2}\right ) \int \arcsin (c x) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {2 a b d x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 d x \sqrt {1-c^2 x^2} \arcsin (c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 c d \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {2 a b d x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 d \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 d x \sqrt {1-c^2 x^2} \arcsin (c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}} \\ \end{align*}
Time = 2.32 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {3 \sqrt {d+c d x} \sqrt {e-c e x} \left (2 a b c x-a^2 \sqrt {1-c^2 x^2}+2 b^2 \sqrt {1-c^2 x^2}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \left (b c x-a \sqrt {1-c^2 x^2}\right ) \arcsin (c x)+3 b \sqrt {d+c d x} \sqrt {e-c e x} \left (a-b \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-3 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )}{3 c e \sqrt {1-c^2 x^2}} \]
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\[\int \frac {\sqrt {c d x +d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {-c e x +e}}d x\]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {\sqrt {c d x + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int \frac {\sqrt {d \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- e \left (c x - 1\right )}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {\sqrt {c d x + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+c d x} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}}{\sqrt {e-c\,e\,x}} \,d x \]
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