Integrand size = 27, antiderivative size = 476 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {100976 b^2 d^3 x}{4002075 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}-\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}-\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {256 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{17325 c^3}+\frac {32 b d^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5775 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{693 c^5}-\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{1155 c^5}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{1617 c^5}-\frac {8 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{297 c^5}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \arcsin (c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{33} d^3 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 \]
-100976/4002075*b^2*d^3*x/c^4-50488/12006225*b^2*d^3*x^3/c^2-12622/6670125 *b^2*d^3*x^5+9410/1120581*b^2*c^2*d^3*x^7-182/29403*b^2*c^4*d^3*x^9+2/1331 *b^2*c^6*d^3*x^11+16/693*b*d^3*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/c^5-4/ 1155*b*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/c^5+2/1617*b*d^3*(-c^2*x^2 +1)^(7/2)*(a+b*arcsin(c*x))/c^5-8/297*b*d^3*(-c^2*x^2+1)^(9/2)*(a+b*arcsin (c*x))/c^5+2/121*b*d^3*(-c^2*x^2+1)^(11/2)*(a+b*arcsin(c*x))/c^5+16/1155*d ^3*x^5*(a+b*arcsin(c*x))^2+8/231*d^3*x^5*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+ 2/33*d^3*x^5*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2+1/11*d^3*x^5*(-c^2*x^2+1)^ 3*(a+b*arcsin(c*x))^2+256/17325*b*d^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2) /c^5+128/17325*b*d^3*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+32/5775* b*d^3*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.63 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {d^3 \left (12006225 a^2 c^5 x^5 \left (-231+495 c^2 x^2-385 c^4 x^4+105 c^6 x^6\right )+6930 a b \sqrt {1-c^2 x^2} \left (-50488-25244 c^2 x^2-18933 c^4 x^4+117625 c^6 x^6-111475 c^8 x^8+33075 c^{10} x^{10}\right )+b^2 \left (349881840 c x+58313640 c^3 x^3+26241138 c^5 x^5-116448750 c^7 x^7+85835750 c^9 x^9-20837250 c^{11} x^{11}\right )+6930 b \left (3465 a c^5 x^5 \left (-231+495 c^2 x^2-385 c^4 x^4+105 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-50488-25244 c^2 x^2-18933 c^4 x^4+117625 c^6 x^6-111475 c^8 x^8+33075 c^{10} x^{10}\right )\right ) \arcsin (c x)+12006225 b^2 c^5 x^5 \left (-231+495 c^2 x^2-385 c^4 x^4+105 c^6 x^6\right ) \arcsin (c x)^2\right )}{13867189875 c^5} \]
-1/13867189875*(d^3*(12006225*a^2*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^ 4 + 105*c^6*x^6) + 6930*a*b*Sqrt[1 - c^2*x^2]*(-50488 - 25244*c^2*x^2 - 18 933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^10) + b^2*(34 9881840*c*x + 58313640*c^3*x^3 + 26241138*c^5*x^5 - 116448750*c^7*x^7 + 85 835750*c^9*x^9 - 20837250*c^11*x^11) + 6930*b*(3465*a*c^5*x^5*(-231 + 495* c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-50488 - 25244 *c^2*x^2 - 18933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^ 10))*ArcSin[c*x] + 12006225*b^2*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]^2))/c^5
Time = 3.19 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.49, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {5202, 27, 5194, 27, 1467, 2009, 5202, 5194, 27, 1467, 2009, 5202, 5138, 5194, 27, 2009, 5210, 15, 5210, 15, 5182, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2}{11} b c d^3 \int x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {6}{11} d \int d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{11} b c d^3 \int x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {6}{11} d^3 \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{11} b c d^3 \left (-b c \int -\frac {\left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )}{693 c^6}dx-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{11} b c d^3 \left (\frac {b \int \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )dx}{693 c^5}-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{11} b c d^3 \left (\frac {b \int \left (-63 c^{10} x^{10}+161 c^8 x^8-113 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{693 c^5}-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {6}{11} d^3 \left (-\frac {2}{9} b c \int x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {4}{9} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c \left (-b c \int -\frac {\left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )}{315 c^6}dx-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c \left (\frac {b \int \left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )dx}{315 c^5}-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c \left (\frac {b \int \left (35 c^8 x^8-50 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{315 c^5}-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (-\frac {2}{7} b c \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{7} \int x^4 (a+b \arcsin (c x))^2dx+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \left (-b c \int -\frac {-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8}{105 c^6}dx-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \left (\frac {b \int \left (-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{105 c^5}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}+\frac {b \int x^4dx}{5 c}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{11} d^3 x^5 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {6}{11} d^3 \left (\frac {1}{9} x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{9} \left (\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )-\frac {2}{9} b c \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )-\frac {2}{11} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6}+\frac {2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {b \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
(d^3*x^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/11 - (2*b*c*d^3*((b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6*x^7)/7 + (161*c^8*x^9)/9 - (63*c ^10*x^11)/11))/(693*c^5) - ((1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^ 6) + (2*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6) - ((1 - c^2*x^2)^ (11/2)*(a + b*ArcSin[c*x]))/(11*c^6)))/11 + (6*d^3*((x^5*(1 - c^2*x^2)^2*( a + b*ArcSin[c*x])^2)/9 - (2*b*c*((b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (50*c^6*x^7)/7 + (35*c^8*x^9)/9))/(315*c^5) - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^6) + (2*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c ^6) - ((1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6)))/9 + (4*((x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7 - (2*b*c*((b*(8*x + (4*c^2*x^3)/3 + (3 *c^4*x^5)/5 - (15*c^6*x^7)/7))/(105*c^5) - ((1 - c^2*x^2)^(3/2)*(a + b*Arc Sin[c*x]))/(3*c^6) + (2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^6) - ((1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6)))/7 + (2*((x^5*(a + b*A rcSin[c*x])^2)/5 - (2*b*c*((b*x^5)/(25*c) - (x^4*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/(5*c^2) + (4*((b*x^3)/(9*c) - (x^2*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/(3*c^2) + (2*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x ]))/c^2))/(3*c^2)))/(5*c^2)))/5))/7))/9))/11
3.2.74.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2 *x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.29 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.41
| method | result | size |
| parts | \(-d^{3} a^{2} \left (\frac {1}{11} c^{6} x^{11}-\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}-\frac {1}{5} x^{5}\right )-\frac {d^{3} b^{2} \left (-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{1925}+\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 {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{1155}+\frac {32 c x}{1155}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3465}-\frac {8 \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}{93555}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{1617}-\frac {2 \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{83853}+\frac {2 \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 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{56595}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{10395}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{5} \sqrt {-c^{2} x^{2}+1}}{121}+\frac {\arcsin \left (c x \right )^{2} \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{693}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{28875}+\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{297}\right )}{c^{5}}-\frac {2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(671\) |
| derivativedivides | \(\frac {-d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}-\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d^{3} b^{2} \left (-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{1925}+\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 {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{1155}+\frac {32 c x}{1155}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3465}-\frac {8 \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}{93555}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{1617}-\frac {2 \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{83853}+\frac {2 \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 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{56595}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{10395}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{5} \sqrt {-c^{2} x^{2}+1}}{121}+\frac {\arcsin \left (c x \right )^{2} \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{693}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{28875}+\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{297}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(672\) |
| default | \(\frac {-d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}-\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d^{3} b^{2} \left (-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{1925}+\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 {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{1155}+\frac {32 c x}{1155}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3465}-\frac {8 \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}{93555}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{1617}-\frac {2 \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{83853}+\frac {2 \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 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{56595}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{10395}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{5} \sqrt {-c^{2} x^{2}+1}}{121}+\frac {\arcsin \left (c x \right )^{2} \left (63 c^{10} x^{10}-385 c^{8} x^{8}+990 c^{6} x^{6}-1386 c^{4} x^{4}+1155 c^{2} x^{2}-693\right ) c x}{693}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{28875}+\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{297}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(672\) |
-d^3*a^2*(1/11*c^6*x^11-1/3*c^4*x^9+3/7*c^2*x^7-1/5*x^5)-d^3*b^2/c^5*(-4/1 925*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)+1/35*arcsin(c*x)^2*(5*c^6 *x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-32/1155*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+ 32/1155*c*x+16/3465*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-8/93555*(35 *c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x+2/1617*arcsin(c*x)*( c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/83853*(63*c^10*x^10-385*c^8*x^8+990*c^6* x^6-1386*c^4*x^4+1155*c^2*x^2-693)*c*x+2/315*arcsin(c*x)^2*(35*c^8*x^8-180 *c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x-2/56595*(5*c^6*x^6-21*c^4*x^4+35 *c^2*x^2-35)*c*x-16/10395*(c^2*x^2-3)*c*x+2/121*arcsin(c*x)*(c^2*x^2-1)^5* (-c^2*x^2+1)^(1/2)+1/693*arcsin(c*x)^2*(63*c^10*x^10-385*c^8*x^8+990*c^6*x ^6-1386*c^4*x^4+1155*c^2*x^2-693)*c*x+4/28875*(3*c^4*x^4-10*c^2*x^2+15)*c* x+8/297*arcsin(c*x)*(c^2*x^2-1)^4*(-c^2*x^2+1)^(1/2))-2*d^3*a*b/c^5*(1/11* arcsin(c*x)*c^11*x^11-1/3*arcsin(c*x)*c^9*x^9+3/7*arcsin(c*x)*c^7*x^7-1/5* arcsin(c*x)*c^5*x^5-91/3267*c^8*x^8*(-c^2*x^2+1)^(1/2)+4705/160083*c^6*x^6 *(-c^2*x^2+1)^(1/2)-6311/1334025*c^4*x^4*(-c^2*x^2+1)^(1/2)-25244/4002075* c^2*x^2*(-c^2*x^2+1)^(1/2)-50488/4002075*(-c^2*x^2+1)^(1/2)+1/121*c^10*x^1 0*(-c^2*x^2+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.87 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {10418625 \, {\left (121 \, a^{2} - 2 \, b^{2}\right )} c^{11} d^{3} x^{11} - 471625 \, {\left (9801 \, a^{2} - 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \, {\left (480249 \, a^{2} - 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} - 2079 \, {\left (1334025 \, a^{2} - 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} + 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \, {\left (105 \, b^{2} c^{11} d^{3} x^{11} - 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} - 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \arcsin \left (c x\right )^{2} + 24012450 \, {\left (105 \, a b c^{11} d^{3} x^{11} - 385 \, a b c^{9} d^{3} x^{9} + 495 \, a b c^{7} d^{3} x^{7} - 231 \, a b c^{5} d^{3} x^{5}\right )} \arcsin \left (c x\right ) + 6930 \, {\left (33075 \, a b c^{10} d^{3} x^{10} - 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} - 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} - 50488 \, a b d^{3} + {\left (33075 \, b^{2} c^{10} d^{3} x^{10} - 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} - 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} - 50488 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \]
-1/13867189875*(10418625*(121*a^2 - 2*b^2)*c^11*d^3*x^11 - 471625*(9801*a^ 2 - 182*b^2)*c^9*d^3*x^9 + 12375*(480249*a^2 - 9410*b^2)*c^7*d^3*x^7 - 207 9*(1334025*a^2 - 12622*b^2)*c^5*d^3*x^5 + 58313640*b^2*c^3*d^3*x^3 + 34988 1840*b^2*c*d^3*x + 12006225*(105*b^2*c^11*d^3*x^11 - 385*b^2*c^9*d^3*x^9 + 495*b^2*c^7*d^3*x^7 - 231*b^2*c^5*d^3*x^5)*arcsin(c*x)^2 + 24012450*(105* a*b*c^11*d^3*x^11 - 385*a*b*c^9*d^3*x^9 + 495*a*b*c^7*d^3*x^7 - 231*a*b*c^ 5*d^3*x^5)*arcsin(c*x) + 6930*(33075*a*b*c^10*d^3*x^10 - 111475*a*b*c^8*d^ 3*x^8 + 117625*a*b*c^6*d^3*x^6 - 18933*a*b*c^4*d^3*x^4 - 25244*a*b*c^2*d^3 *x^2 - 50488*a*b*d^3 + (33075*b^2*c^10*d^3*x^10 - 111475*b^2*c^8*d^3*x^8 + 117625*b^2*c^6*d^3*x^6 - 18933*b^2*c^4*d^3*x^4 - 25244*b^2*c^2*d^3*x^2 - 50488*b^2*d^3)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^5
Time = 2.98 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.47 \[ \int x^4 \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^{11}}{11} + \frac {a^{2} c^{4} d^{3} x^{9}}{3} - \frac {3 a^{2} c^{2} d^{3} x^{7}}{7} + \frac {a^{2} d^{3} x^{5}}{5} - \frac {2 a b c^{6} d^{3} x^{11} \operatorname {asin}{\left (c x \right )}}{11} - \frac {2 a b c^{5} d^{3} x^{10} \sqrt {- c^{2} x^{2} + 1}}{121} + \frac {2 a b c^{4} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{3} + \frac {182 a b c^{3} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{3267} - \frac {6 a b c^{2} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {9410 a b c d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{160083} + \frac {2 a b d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {12622 a b d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1334025 c} + \frac {50488 a b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{4002075 c^{3}} + \frac {100976 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{4002075 c^{5}} - \frac {b^{2} c^{6} d^{3} x^{11} \operatorname {asin}^{2}{\left (c x \right )}}{11} + \frac {2 b^{2} c^{6} d^{3} x^{11}}{1331} - \frac {2 b^{2} c^{5} d^{3} x^{10} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{121} + \frac {b^{2} c^{4} d^{3} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {182 b^{2} c^{4} d^{3} x^{9}}{29403} + \frac {182 b^{2} c^{3} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3267} - \frac {3 b^{2} c^{2} d^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {9410 b^{2} c^{2} d^{3} x^{7}}{1120581} - \frac {9410 b^{2} c d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{160083} + \frac {b^{2} d^{3} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {12622 b^{2} d^{3} x^{5}}{6670125} + \frac {12622 b^{2} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1334025 c} - \frac {50488 b^{2} d^{3} x^{3}}{12006225 c^{2}} + \frac {50488 b^{2} d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{4002075 c^{3}} - \frac {100976 b^{2} d^{3} x}{4002075 c^{4}} + \frac {100976 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**6*d**3*x**11/11 + a**2*c**4*d**3*x**9/3 - 3*a**2*c**2* d**3*x**7/7 + a**2*d**3*x**5/5 - 2*a*b*c**6*d**3*x**11*asin(c*x)/11 - 2*a* b*c**5*d**3*x**10*sqrt(-c**2*x**2 + 1)/121 + 2*a*b*c**4*d**3*x**9*asin(c*x )/3 + 182*a*b*c**3*d**3*x**8*sqrt(-c**2*x**2 + 1)/3267 - 6*a*b*c**2*d**3*x **7*asin(c*x)/7 - 9410*a*b*c*d**3*x**6*sqrt(-c**2*x**2 + 1)/160083 + 2*a*b *d**3*x**5*asin(c*x)/5 + 12622*a*b*d**3*x**4*sqrt(-c**2*x**2 + 1)/(1334025 *c) + 50488*a*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(4002075*c**3) + 100976*a*b *d**3*sqrt(-c**2*x**2 + 1)/(4002075*c**5) - b**2*c**6*d**3*x**11*asin(c*x) **2/11 + 2*b**2*c**6*d**3*x**11/1331 - 2*b**2*c**5*d**3*x**10*sqrt(-c**2*x **2 + 1)*asin(c*x)/121 + b**2*c**4*d**3*x**9*asin(c*x)**2/3 - 182*b**2*c** 4*d**3*x**9/29403 + 182*b**2*c**3*d**3*x**8*sqrt(-c**2*x**2 + 1)*asin(c*x) /3267 - 3*b**2*c**2*d**3*x**7*asin(c*x)**2/7 + 9410*b**2*c**2*d**3*x**7/11 20581 - 9410*b**2*c*d**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/160083 + b**2 *d**3*x**5*asin(c*x)**2/5 - 12622*b**2*d**3*x**5/6670125 + 12622*b**2*d**3 *x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(1334025*c) - 50488*b**2*d**3*x**3/(1 2006225*c**2) + 50488*b**2*d**3*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(40020 75*c**3) - 100976*b**2*d**3*x/(4002075*c**4) + 100976*b**2*d**3*sqrt(-c**2 *x**2 + 1)*asin(c*x)/(4002075*c**5), Ne(c, 0)), (a**2*d**3*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (421) = 842\).
Time = 0.32 (sec) , antiderivative size = 1141, normalized size of antiderivative = 2.40 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \]
-1/11*b^2*c^6*d^3*x^11*arcsin(c*x)^2 - 1/11*a^2*c^6*d^3*x^11 + 1/3*b^2*c^4 *d^3*x^9*arcsin(c*x)^2 + 1/3*a^2*c^4*d^3*x^9 - 3/7*b^2*c^2*d^3*x^7*arcsin( c*x)^2 - 3/7*a^2*c^2*d^3*x^7 - 2/7623*(693*x^11*arcsin(c*x) + (63*sqrt(-c^ 2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(-c^2*x^2 + 1 )*x^6/c^6 + 96*sqrt(-c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^2/c^1 0 + 256*sqrt(-c^2*x^2 + 1)/c^12)*c)*a*b*c^6*d^3 - 2/26413695*(3465*(63*sqr t(-c^2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(-c^2*x^ 2 + 1)*x^6/c^6 + 96*sqrt(-c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^ 2/c^10 + 256*sqrt(-c^2*x^2 + 1)/c^12)*c*arcsin(c*x) - (19845*c^10*x^11 + 2 6950*c^8*x^9 + 39600*c^6*x^7 + 66528*c^4*x^5 + 147840*c^2*x^3 + 887040*x)/ c^10)*b^2*c^6*d^3 + 1/5*b^2*d^3*x^5*arcsin(c*x)^2 + 2/945*(315*x^9*arcsin( c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48 *sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^ 2*x^2 + 1)/c^10)*c)*a*b*c^4*d^3 + 2/297675*(315*(35*sqrt(-c^2*x^2 + 1)*x^8 /c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64* sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c*arcsin(c*x) - (1225*c^8*x^9 + 1800*c^6*x^7 + 3024*c^4*x^5 + 6720*c^2*x^3 + 40320*x)/c^8) *b^2*c^4*d^3 + 1/5*a^2*d^3*x^5 - 6/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2* x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2 /c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^2*d^3 - 2/8575*(105*(5*sqrt(...
Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (421) = 842\).
Time = 0.34 (sec) , antiderivative size = 865, normalized size of antiderivative = 1.82 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\text {Too large to display} \]
-1/11*a^2*c^6*d^3*x^11 + 1/3*a^2*c^4*d^3*x^9 - 3/7*a^2*c^2*d^3*x^7 + 1/5*a ^2*d^3*x^5 - 1/11*(c^2*x^2 - 1)^5*b^2*d^3*x*arcsin(c*x)^2/c^4 - 2/11*(c^2* x^2 - 1)^5*a*b*d^3*x*arcsin(c*x)/c^4 - 4/33*(c^2*x^2 - 1)^4*b^2*d^3*x*arcs in(c*x)^2/c^4 + 2/1331*(c^2*x^2 - 1)^5*b^2*d^3*x/c^4 - 8/33*(c^2*x^2 - 1)^ 4*a*b*d^3*x*arcsin(c*x)/c^4 - 1/231*(c^2*x^2 - 1)^3*b^2*d^3*x*arcsin(c*x)^ 2/c^4 - 2/121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 + 428/323433*(c^2*x^2 - 1)^4*b^2*d^3*x/c^4 - 2/231*(c^2*x^2 - 1)^3*a*b*d^3* x*arcsin(c*x)/c^4 + 2/385*(c^2*x^2 - 1)^2*b^2*d^3*x*arcsin(c*x)^2/c^4 - 2/ 121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 - 8/297*(c^2*x^2 - 1)^4 *sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 - 148174/110937519*(c^2*x^2 - 1)^3*b^2*d^3*x/c^4 + 4/385*(c^2*x^2 - 1)^2*a*b*d^3*x*arcsin(c*x)/c^4 - 8/1 155*(c^2*x^2 - 1)*b^2*d^3*x*arcsin(c*x)^2/c^4 - 8/297*(c^2*x^2 - 1)^4*sqrt (-c^2*x^2 + 1)*a*b*d^3/c^5 - 2/1617*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2 *d^3*arcsin(c*x)/c^5 + 5487704/4622396625*(c^2*x^2 - 1)^2*b^2*d^3*x/c^4 - 16/1155*(c^2*x^2 - 1)*a*b*d^3*x*arcsin(c*x)/c^4 + 16/1155*b^2*d^3*x*arcsin (c*x)^2/c^4 - 2/1617*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 + 4/19 25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^5 - 606416/138 67189875*(c^2*x^2 - 1)*b^2*d^3*x/c^4 + 32/1155*a*b*d^3*x*arcsin(c*x)/c^4 + 4/1925*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^5 + 16/3465*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*arcsin(c*x)/c^5 - 382986368/13867189875*b^2*d^3*x/c...
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]