Integrand size = 30, antiderivative size = 96 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4763, 4745, 266} \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rule 266
Rule 4745
Rule 4763
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \\ & = \frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \\ & = \frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {\sqrt {d+c d x} \left (2 a c x+2 b c x \arcsin (c x)+b \sqrt {1-c^2 x^2} \log (-f (1+c x))+b \sqrt {1-c^2 x^2} \log (f-c f x)\right )}{2 c d^2 f (1+c x) \sqrt {f-c f x}} \]
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\[\int \frac {a +b \arcsin \left (c x \right )}{\left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (- f \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {b x \arcsin \left (c x\right )}{\sqrt {-c^{2} d f x^{2} + d f} d f} + \frac {a x}{\sqrt {-c^{2} d f x^{2} + d f} d f} - \frac {b \sqrt {\frac {1}{d f}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, c d f} \]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (f-c\,f\,x\right )}^{3/2}} \,d x \]
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