Integrand size = 27, antiderivative size = 391 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {10516 b^2 d^3 x}{99225 c^2}-\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c^3}+\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 \]
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Time = 0.52 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4787, 4723, 4795, 4767, 8, 30, 272, 45, 4779, 12, 380} \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {64 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {2}{729} b^2 c^6 d^3 x^9-\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {10516 b^2 d^3 x}{99225 c^2}-\frac {5258 b^2 d^3 x^3}{297675} \]
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Rule 8
Rule 12
Rule 30
Rule 45
Rule 272
Rule 380
Rule 4723
Rule 4767
Rule 4779
Rule 4787
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {1}{3} (2 d) \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx-\frac {1}{9} \left (2 b c d^3\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx \\ & = \frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{63 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {1}{21} \left (8 d^2\right ) \int x^2 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx-\frac {1}{21} \left (4 b c d^3\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx+\frac {1}{9} \left (2 b^2 c^2 d^3\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx \\ & = \frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{105 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {1}{105} \left (16 d^3\right ) \int x^2 (a+b \arcsin (c x))^2 \, dx+\frac {\left (2 b^2 d^3\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{567 c^2}-\frac {1}{105} \left (16 b c d^3\right ) \int x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx+\frac {1}{21} \left (4 b^2 c^2 d^3\right ) \int \frac {\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx \\ & = \frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {\left (2 b^2 d^3\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{567 c^2}+\frac {\left (4 b^2 d^3\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{735 c^2}-\frac {1}{315} \left (32 b c d^3\right ) \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{105} \left (16 b^2 c^2 d^3\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx \\ & = -\frac {4 b^2 d^3 x}{567 c^2}-\frac {2 b^2 d^3 x^3}{1701}+\frac {2}{189} b^2 c^2 d^3 x^5-\frac {38 b^2 c^4 d^3 x^7}{3969}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {1}{945} \left (32 b^2 d^3\right ) \int x^2 \, dx+\frac {\left (4 b^2 d^3\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{735 c^2}+\frac {\left (16 b^2 d^3\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{1575 c^2}-\frac {\left (64 b d^3\right ) \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{945 c} \\ & = -\frac {3796 b^2 d^3 x}{99225 c^2}-\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c^3}+\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {\left (64 b^2 d^3\right ) \int 1 \, dx}{945 c^2} \\ & = -\frac {10516 b^2 d^3 x}{99225 c^2}-\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c^3}+\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.71 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {d^3 \left (99225 a^2 c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+630 a b \sqrt {1-c^2 x^2} \left (-5258-2629 c^2 x^2+6297 c^4 x^4-4675 c^6 x^6+1225 c^8 x^8\right )+b^2 \left (3312540 c x+552090 c^3 x^3-793422 c^5 x^5+420750 c^7 x^7-85750 c^9 x^9\right )+630 b \left (315 a c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-5258-2629 c^2 x^2+6297 c^4 x^4-4675 c^6 x^6+1225 c^8 x^8\right )\right ) \arcsin (c x)+99225 b^2 c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right ) \arcsin (c x)^2\right )}{31255875 c^3} \]
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Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.34
method | result | size |
parts | \(-d^{3} a^{2} \left (\frac {1}{9} c^{6} x^{9}-\frac {3}{7} c^{4} x^{7}+\frac {3}{5} c^{2} x^{5}-\frac {1}{3} x^{3}\right )-\frac {d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )}{c^{3}}-\frac {2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(524\) |
derivativedivides | \(\frac {-d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(525\) |
default | \(\frac {-d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(525\) |
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Time = 0.27 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.95 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {42875 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{3} x^{9} - 1125 \, {\left (11907 \, a^{2} - 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} - 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} - 105 \, {\left (99225 \, a^{2} - 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} + 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} - 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} - 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right )^{2} + 198450 \, {\left (35 \, a b c^{9} d^{3} x^{9} - 135 \, a b c^{7} d^{3} x^{7} + 189 \, a b c^{5} d^{3} x^{5} - 105 \, a b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + 630 \, {\left (1225 \, a b c^{8} d^{3} x^{8} - 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} - 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3} + {\left (1225 \, b^{2} c^{8} d^{3} x^{8} - 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} - 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{31255875 \, c^{3}} \]
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Time = 1.63 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.60 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{6} d^{3} x^{9}}{9} + \frac {3 a^{2} c^{4} d^{3} x^{7}}{7} - \frac {3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac {a^{2} d^{3} x^{3}}{3} - \frac {2 a b c^{6} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{9} - \frac {2 a b c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} + \frac {6 a b c^{4} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {374 a b c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} - \frac {6 a b c^{2} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {4198 a b c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075} + \frac {2 a b d^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {5258 a b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c} + \frac {10516 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {b^{2} c^{6} d^{3} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac {2 b^{2} c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} - \frac {374 b^{2} c^{4} d^{3} x^{7}}{27783} + \frac {374 b^{2} c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3969} - \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} + \frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac {4198 b^{2} c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {5258 b^{2} d^{3} x^{3}}{297675} + \frac {5258 b^{2} d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c} - \frac {10516 b^{2} d^{3} x}{99225 c^{2}} + \frac {10516 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 946 vs. \(2 (346) = 692\).
Time = 0.32 (sec) , antiderivative size = 946, normalized size of antiderivative = 2.42 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{9} \, b^{2} c^{6} d^{3} x^{9} \arcsin \left (c x\right )^{2} - \frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{4} d^{3} x^{7} \arcsin \left (c x\right )^{2} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, b^{2} c^{2} d^{3} x^{5} \arcsin \left (c x\right )^{2} - \frac {2}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} a b c^{6} d^{3} - \frac {2}{893025} \, {\left (315 \, {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c \arcsin \left (c x\right ) - \frac {1225 \, c^{8} x^{9} + 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} + 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{6} d^{3} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {6}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{3} + \frac {2}{8575} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{3} + \frac {1}{3} \, b^{2} d^{3} x^{3} \arcsin \left (c x\right )^{2} - \frac {2}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d^{3} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{3} + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (346) = 692\).
Time = 0.34 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.83 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{9 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x}{729 \, c^{2}} + \frac {1}{3} \, a^{2} d^{3} x^{3} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{3} x \arcsin \left (c x\right )}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{105 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{81 \, c^{3}} - \frac {622 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x}{250047 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{3} x \arcsin \left (c x\right )}{105 \, c^{2}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{81 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{441 \, c^{3}} + \frac {15224 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x}{10418625 \, c^{2}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )} a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {16 \, b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{441 \, c^{3}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{525 \, c^{3}} + \frac {115504 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {32 \, a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{525 \, c^{3}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} \arcsin \left (c x\right )}{945 \, c^{3}} - \frac {3406208 \, b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3}}{945 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{315 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{315 \, c^{3}} \]
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Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]
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