Integrand size = 30, antiderivative size = 99 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b f \left (1-c^2 x^2\right )^{3/2} \log (1+c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \]
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Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4763, 651, 4845, 12, 641, 31} \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\frac {b f \left (1-c^2 x^2\right )^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rule 12
Rule 31
Rule 641
Rule 651
Rule 4763
Rule 4845
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(f-c f x) (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 {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (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 {f (1-c x)}{c \left (1-c^2 x^2\right )} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {\left (b f \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {1-c x}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {\left (b f \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {1}{1+c x} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b f \left (1-c^2 x^2\right )^{3/2} \log (1+c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \\ \end{align*}
Time = 0.82 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\frac {\sqrt {d+c d x} \left (a (-1+c x)+b (-1+c x) \arcsin (c x)+b \sqrt {1-c^2 x^2} \log (-f (1+c x))\right )}{c d^2 (1+c x) \sqrt {f-c f x}} \]
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\[\int \frac {a +b \arcsin \left (c x \right )}{\left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c f x +f}}d x\]
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Time = 0.31 (sec) , antiderivative size = 348, normalized size of antiderivative = 3.52 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\left [\frac {{\left (b c x + b\right )} \sqrt {d f} \log \left (\frac {c^{6} d f x^{6} + 4 \, c^{5} d f x^{5} + 5 \, c^{4} d f x^{4} - 4 \, c^{2} d f x^{2} - 4 \, c d f x - {\left (c^{4} x^{4} + 4 \, c^{3} x^{3} + 6 \, c^{2} x^{2} + 4 \, c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {d f} - 2 \, d f}{c^{4} x^{4} + 2 \, c^{3} x^{3} - 2 \, c x - 1}\right ) - 2 \, \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{2 \, {\left (c^{2} d^{2} f x + c d^{2} f\right )}}, \frac {{\left (b c x + b\right )} \sqrt {-d f} \arctan \left (\frac {{\left (c^{2} x^{2} + 2 \, c x + 2\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {-d f}}{c^{4} d f x^{4} + 2 \, c^{3} d f x^{3} - c^{2} d f x^{2} - 2 \, c d f x}\right ) - \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{2} d^{2} f x + c d^{2} f}\right ] \]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \sqrt {- f \left (c x - 1\right )}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=-\frac {\sqrt {-c^{2} d f x^{2} + d f} b \arcsin \left (c x\right )}{c^{2} d^{2} f x + c d^{2} f} - \frac {\sqrt {-c^{2} d f x^{2} + d f} a}{c^{2} d^{2} f x + c d^{2} f} + \frac {b \log \left (c x + 1\right )}{c d^{\frac {3}{2}} \sqrt {f}} \]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {3}{2}} \sqrt {-c f x + f}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} \sqrt {f-c f x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+c\,d\,x\right )}^{3/2}\,\sqrt {f-c\,f\,x}} \,d x \]
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