Integrand size = 35, antiderivative size = 509 \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {7 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \]
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Time = 0.70 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {4823, 4787, 4783, 4795, 4737, 4723, 327, 222, 14, 4777, 12, 470} \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {b d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} d e x^3 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2-\frac {d e x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {d e \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {7 b^2 d e \arcsin (c x) \sqrt {c d x+d} \sqrt {e-c e x}}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {c d x+d} \sqrt {e-c e x}-\frac {7 b^2 d e x \sqrt {c d x+d} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {c d x+d} \sqrt {e-c e x}}{1728} \]
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Rule 12
Rule 14
Rule 222
Rule 327
Rule 470
Rule 4723
Rule 4737
Rule 4777
Rule 4783
Rule 4787
Rule 4795
Rule 4823
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 (a+b \arcsin (c x)) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x (a+b \arcsin (c x)) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{64} b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}} \\ & = \frac {b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{128 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {7 b^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}}{1152 c^2}-\frac {43 b^2 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x}}{1728}+\frac {1}{108} b^2 c^2 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {7 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d e x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d e x^6 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{6} d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 2.26 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.89 \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {288 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-864 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-12 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (-18 b \cos (2 \arcsin (c x))+9 b \cos (4 \arcsin (c x))+2 b \cos (6 \arcsin (c x))-36 a \sin (2 \arcsin (c x))+36 a \sin (4 \arcsin (c x))+12 a \sin (6 \arcsin (c x)))-72 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (-12 a-3 b \sin (2 \arcsin (c x))+3 b \sin (4 \arcsin (c x))+b \sin (6 \arcsin (c x)))+d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-864 a^2 c x \sqrt {1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+216 a b \cos (2 \arcsin (c x))-108 a b \cos (4 \arcsin (c x))-24 a b \cos (6 \arcsin (c x))-108 b^2 \sin (2 \arcsin (c x))+27 b^2 \sin (4 \arcsin (c x))+4 b^2 \sin (6 \arcsin (c x))\right )}{13824 c^3 \sqrt {1-c^2 x^2}} \]
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\[\int x^{2} \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
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\[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
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Exception generated. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]
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