Integrand size = 30, antiderivative size = 255 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=-\frac {b f \left (1-c^2 x^2\right )^{5/2}}{6 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \text {arctanh}(c x)}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}} \]
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Time = 0.17 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4763, 653, 197, 4845, 641, 46, 213, 266} \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \text {arctanh}(c x)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b f \left (1-c^2 x^2\right )^{5/2}}{6 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
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Rule 46
Rule 197
Rule 213
Rule 266
Rule 641
Rule 653
Rule 4763
Rule 4845
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(f-c f x) (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {f (1-c x)}{3 c \left (1-c^2 x^2\right )^2}+\frac {2 f x}{3 \left (1-c^2 x^2\right )}\right ) \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1-c x}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (2 b c f \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {b f \left (1-c^2 x^2\right )^{5/2}}{6 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b f \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{6 (d+c d x)^{5/2} (f-c f x)^{5/2}} \\ & = -\frac {b f \left (1-c^2 x^2\right )^{5/2}}{6 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {f (1-c x) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \text {arctanh}(c x)}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b f \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}} \\ \end{align*}
Time = 1.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\frac {\sqrt {d+c d x} \left (-4 a+8 a c x+8 a c^2 x^2-2 b \sqrt {1-c^2 x^2}+4 b \left (-1+2 c x+2 c^2 x^2\right ) \arcsin (c x)+5 b (1+c x) \sqrt {1-c^2 x^2} \log (-f (1+c x))+3 b \sqrt {1-c^2 x^2} \log (f-c f x)+3 b c x \sqrt {1-c^2 x^2} \log (f-c f x)\right )}{12 c d^3 f (1+c x)^2 \sqrt {f-c f x}} \]
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\[\int \frac {a +b \arcsin \left (c x \right )}{\left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=-\frac {1}{12} \, b c {\left (\frac {2 \, \sqrt {d} \sqrt {f}}{c^{3} d^{3} f^{2} x + c^{2} d^{3} f^{2}} - \frac {5 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}} f^{\frac {3}{2}}} - \frac {3 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}} f^{\frac {3}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {1}{\sqrt {-c^{2} d f x^{2} + d f} c^{2} d^{2} f x + \sqrt {-c^{2} d f x^{2} + d f} c d^{2} f} - \frac {2 \, x}{\sqrt {-c^{2} d f x^{2} + d f} d^{2} f}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {1}{\sqrt {-c^{2} d f x^{2} + d f} c^{2} d^{2} f x + \sqrt {-c^{2} d f x^{2} + d f} c d^{2} f} - \frac {2 \, x}{\sqrt {-c^{2} d f x^{2} + d f} d^{2} f}\right )} \]
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\[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+c\,d\,x\right )}^{5/2}\,{\left (f-c\,f\,x\right )}^{3/2}} \,d x \]
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