Integrand size = 32, antiderivative size = 697 \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {8 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}}{225 c}-\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {16 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac {15 b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {2 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right )}{125 c}+\frac {9 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2} \arcsin (c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c e x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}} \]
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Time = 0.52 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {4763, 4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1261, 712} \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {3 b c e x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}-\frac {2 b e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {e (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {e \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{5 c}+\frac {b e \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 c}+\frac {4 b c^2 e x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} e x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {9 b^2 e \arcsin (c x) (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {15 b^2 e x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {2 b^2 e \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}-\frac {16 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac {1}{32} b^2 e x (c d x+d)^{3/2} (e-c e x)^{3/2}-\frac {8 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \]
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Rule 12
Rule 200
Rule 201
Rule 222
Rule 327
Rule 712
Rule 1261
Rule 4723
Rule 4737
Rule 4739
Rule 4741
Rule 4743
Rule 4763
Rule 4767
Rule 4847
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int (e-c e x) \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (e \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-c e x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {\left (e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}-\frac {\left (c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {\left (3 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (2 b e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x)) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {\left (3 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x (a+b \arcsin (c x)) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (2 b^2 c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c e x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (2 b^2 c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c e x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (b^2 c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2} \arcsin (c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c e x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (b^2 c e (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \left (1-c^2 x^2\right )^{3/2}} \\ & = -\frac {8 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}}{225 c}-\frac {1}{32} b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {16 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac {15 b^2 e x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {2 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right )}{125 c}+\frac {9 b^2 e (d+c d x)^{3/2} (e-c e x)^{3/2} \arcsin (c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c e x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^2 e x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^4 e x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b e (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}+\frac {1}{4} e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2+\frac {3 e x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{5 c}+\frac {e (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 4.54 (sec) , antiderivative size = 574, normalized size of antiderivative = 0.82 \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d e^2 \left (36000 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-108000 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+1800 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (10 b \cos (3 \arcsin (c x))+2 b \cos (5 \arcsin (c x))+5 \left (12 a+4 b \sqrt {1-c^2 x^2}+8 b \sin (2 \arcsin (c x))+b \sin (4 \arcsin (c x))\right )\right )+\sqrt {d+c d x} \sqrt {e-c e x} \left (72000 a b \cos (2 \arcsin (c x))-4000 b^2 \cos (3 \arcsin (c x))+4500 a b \cos (4 \arcsin (c x))-288 b^2 \cos (5 \arcsin (c x))-15 \left (4800 b^2 \sqrt {1-c^2 x^2}+512 a b c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-480 a^2 \sqrt {1-c^2 x^2} \left (8+25 c x-16 c^2 x^2-10 c^3 x^3+8 c^4 x^4\right )+2400 b^2 \sin (2 \arcsin (c x))+75 b^2 \sin (4 \arcsin (c x))\right )\right )+60 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (1200 b \cos (2 \arcsin (c x))+75 b \cos (4 \arcsin (c x))+4 \left (-300 b c x+480 a \sqrt {1-c^2 x^2}-960 a c^2 x^2 \sqrt {1-c^2 x^2}+480 a c^4 x^4 \sqrt {1-c^2 x^2}+600 a \sin (2 \arcsin (c x))-50 b \sin (3 \arcsin (c x))+75 a \sin (4 \arcsin (c x))-6 b \sin (5 \arcsin (c x))\right )\right )\right )}{288000 c \sqrt {1-c^2 x^2}} \]
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\[\int \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
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\[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
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Exception generated. \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \]
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\[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (d+c d x)^{3/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{5/2} \,d x \]
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