Integrand size = 30, antiderivative size = 414 \[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {b f x (d+c d x)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {5 b c f x^2 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^2 f x^3 (d+c d x)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 f x^4 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c^4 f x^5 (d+c d x)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))+\frac {3 f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )}+\frac {f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{5 c}+\frac {3 f (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))^2}{16 b c \left (1-c^2 x^2\right )^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4763, 4847, 4743, 4741, 4737, 30, 14, 4767, 200} \[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {3 f x (c d x+d)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )}+\frac {3 f (c d x+d)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {f \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))}{5 c}+\frac {1}{4} f x (c d x+d)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))-\frac {5 b c f x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {b f x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^2 f x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c^4 f x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 f x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}} \]
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Rule 14
Rule 30
Rule 200
Rule 4737
Rule 4741
Rule 4743
Rule 4763
Rule 4767
Rule 4847
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int (f-c f x) \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (f \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-c f x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {\left (f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{\left (1-c^2 x^2\right )^{3/2}}-\frac {\left (c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))+\frac {f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{5 c}+\frac {\left (3 f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))+\frac {3 f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )}+\frac {f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{5 c}+\frac {\left (3 f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c f (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \, dx}{8 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {b f x (d+c d x)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {5 b c f x^2 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^2 f x^3 (d+c d x)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 f x^4 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c^4 f x^5 (d+c d x)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))+\frac {3 f x (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )}+\frac {f (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{5 c}+\frac {3 f (d+c d x)^{3/2} (f-c f x)^{3/2} (a+b \arcsin (c x))^2}{16 b c \left (1-c^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 2.64 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.74 \[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d f^2 \left (1800 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x)^2-3600 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (-128 b c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+240 a \sqrt {1-c^2 x^2} \left (8+25 c x-16 c^2 x^2-10 c^3 x^3+8 c^4 x^4\right )+1200 b \cos (2 \arcsin (c x))+75 b \cos (4 \arcsin (c x))\right )+60 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x) \left (32 \left (1-c^2 x^2\right )^{5/2}+40 \sin (2 \arcsin (c x))+5 \sin (4 \arcsin (c x))\right )\right )}{9600 c \sqrt {1-c^2 x^2}} \]
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\[\int \left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )d x\]
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\[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
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\[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (d+c d x)^{3/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \]
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