Integrand size = 26, antiderivative size = 382 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \] Output:
-10516/99225*b^2*d^3*x/c^2+5258/297675*b^2*d^3*x^3+4198/165375*b^2*c^2*d^3 *x^5+374/27783*b^2*c^4*d^3*x^7+2/729*b^2*c^6*d^3*x^9+64/945*b*d^3*(c^2*x^2 +1)^(1/2)*(a+b*arcsinh(c*x))/c^3-32/945*b*d^3*x^2*(c^2*x^2+1)^(1/2)*(a+b*a rcsinh(c*x))/c+16/315*b*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^3+4/525 *b*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^3+2/441*b*d^3*(c^2*x^2+1)^(7 /2)*(a+b*arcsinh(c*x))/c^3-2/81*b*d^3*(c^2*x^2+1)^(9/2)*(a+b*arcsinh(c*x)) /c^3+16/315*d^3*x^3*(a+b*arcsinh(c*x))^2+8/105*d^3*x^3*(c^2*x^2+1)*(a+b*ar csinh(c*x))^2+2/21*d^3*x^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/9*d^3*x^3* (c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2
Time = 1.28 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.72 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (99225 a^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )-630 a b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )+b^2 \left (-3312540 c x+552090 c^3 x^3+793422 c^5 x^5+420750 c^7 x^7+85750 c^9 x^9\right )-630 b \left (-315 a c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )\right ) \text {arcsinh}(c x)+99225 b^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{31255875 c^3} \] Input:
Integrate[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(99225*a^2*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) - 6 30*a*b*Sqrt[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x ^6 + 1225*c^8*x^8) + b^2*(-3312540*c*x + 552090*c^3*x^3 + 793422*c^5*x^5 + 420750*c^7*x^7 + 85750*c^9*x^9) - 630*b*(-315*a*c^3*x^3*(105 + 189*c^2*x^ 2 + 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSinh[c*x] + 99225*b^2*c^ 3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6)*ArcSinh[c*x]^2))/(312 55875*c^3)
Time = 3.86 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.38, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {6223, 27, 6219, 27, 290, 2009, 6223, 6219, 27, 290, 2009, 6223, 6191, 6219, 27, 2009, 6227, 15, 6213, 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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2}{9} b c d^3 \int x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} d \int d^2 x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^3 \int x^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^3 \left (-b c \int -\frac {\left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3}{63 c^4}dx+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^3 \left (\frac {b \int \left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3dx}{63 c^3}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^3 \left (\frac {b \int \left (-7 c^8 x^8-19 c^6 x^6-15 c^4 x^4-c^2 x^2+2\right )dx}{63 c^3}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c \int x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c \left (-b c \int -\frac {\left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{35 c^4}dx+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c \left (\frac {b \int \left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2dx}{35 c^3}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c \left (\frac {b \int \left (-5 c^6 x^6-8 c^4 x^4-c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (-\frac {2}{5} b c \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{5} \int x^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \left (-b c \int -\frac {-3 c^4 x^4-c^2 x^2+2}{15 c^4}dx+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \left (\frac {b \int \left (-3 c^4 x^4-c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d^3 \left (\frac {1}{7} x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{7} \left (\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )-\frac {2}{9} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
Input:
Int[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*x^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/9 - (2*b*c*d^3*((b*(2*x - (c^2*x^3)/3 - 3*c^4*x^5 - (19*c^6*x^7)/7 - (7*c^8*x^9)/9))/(63*c^3) - ((1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4) + ((1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4)))/9 + (2*d^3*((x^3*(1 + c^2*x^2)^2*(a + b*ArcSi nh[c*x])^2)/7 - (2*b*c*((b*(2*x - (c^2*x^3)/3 - (8*c^4*x^5)/5 - (5*c^6*x^7 )/7))/(35*c^3) - ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4) + ((1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4)))/7 + (4*((x^3*(1 + c^2*x^2 )*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*((b*(2*x - (c^2*x^3)/3 - (3*c^4*x^5)/ 5))/(15*c^3) - ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^4) + ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4)))/5 + (2*((x^3*(a + b*ArcSinh [c*x])^2)/3 - (2*b*c*(-1/9*(b*x^3)/c + (x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSi nh[c*x]))/(3*c^2) - (2*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x ]))/c^2))/(3*c^2)))/3))/5))/7))/3
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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[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] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(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*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcSinh[(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*Arc Sinh[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*ArcSinh[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Simp[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*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 3.16 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {1}{9} c^{6} x^{9}+\frac {3}{7} c^{4} x^{7}+\frac {3}{5} x^{5} c^{2}+\frac {1}{3} x^{3}\right )+\frac {d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 x c}{31255875}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 x c \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )}{c^{3}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{3}}\) | \(437\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 x c}{31255875}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 x c \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{3}}\) | \(438\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 x c}{31255875}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 x c \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {2629 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2099 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {187 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{3}}\) | \(438\) |
| orering | \(\frac {\left (9303875 c^{12} x^{12}+47172500 c^{10} x^{10}+95052594 c^{8} x^{8}+88615068 c^{6} x^{6}-86474829 c^{4} x^{4}-59625720 c^{2} x^{2}-13250160\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{31255875 x \,c^{4} \left (c^{2} x^{2}+1\right )^{4}}-\frac {\left (1029000 c^{10} x^{10}+5185625 c^{8} x^{8}+10248345 c^{6} x^{6}+8539209 c^{4} x^{4}-24568005 c^{2} x^{2}-8281350\right ) \left (2 x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+6 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{31255875 x^{2} c^{4} \left (c^{2} x^{2}+1\right )^{3}}+\frac {\left (42875 c^{8} x^{8}+210375 c^{6} x^{6}+396711 c^{4} x^{4}+276045 c^{2} x^{2}-1656270\right ) \left (2 \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+30 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {8 x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+24 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {24 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{31255875 x \,c^{4} \left (c^{2} x^{2}+1\right )^{2}}\) | \(540\) |
Input:
int(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
d^3*a^2*(1/9*c^6*x^9+3/7*c^4*x^7+3/5*x^5*c^2+1/3*x^3)+d^3*b^2/c^3*(1/9*arc sinh(x*c)^2*x*c*(c^2*x^2+1)^4-16/315*arcsinh(x*c)^2*x*c-1/63*arcsinh(x*c)^ 2*x*c*(c^2*x^2+1)^3-2/105*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^2-8/315*arcsinh(x *c)^2*x*c*(c^2*x^2+1)-2/81*arcsinh(x*c)*(c^2*x^2+1)^(9/2)-3406208/31255875 *x*c+2/729*x*c*(c^2*x^2+1)^4+622/250047*x*c*(c^2*x^2+1)^3+15224/10418625*x *c*(c^2*x^2+1)^2-115504/31255875*x*c*(c^2*x^2+1)+32/315*arcsinh(x*c)*(c^2* x^2+1)^(1/2)+2/441*arcsinh(x*c)*(c^2*x^2+1)^(7/2)+4/525*arcsinh(x*c)*(c^2* x^2+1)^(5/2)+16/945*arcsinh(x*c)*(c^2*x^2+1)^(3/2))+2*d^3*a*b/c^3*(1/9*arc sinh(x*c)*x^9*c^9+3/7*arcsinh(x*c)*c^7*x^7+3/5*arcsinh(x*c)*x^5*c^5+1/3*ar csinh(x*c)*x^3*c^3-2629/99225*x^2*c^2*(c^2*x^2+1)^(1/2)+5258/99225*(c^2*x^ 2+1)^(1/2)-2099/33075*x^4*c^4*(c^2*x^2+1)^(1/2)-187/3969*x^6*c^6*(c^2*x^2+ 1)^(1/2)-1/81*(c^2*x^2+1)^(1/2)*x^8*c^8)
Time = 0.10 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.05 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {42875 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{3} x^{9} + 1125 \, {\left (11907 \, a^{2} + 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} + 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} + 105 \, {\left (99225 \, a^{2} + 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} - 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} + 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} + 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 630 \, {\left (11025 \, a b c^{9} d^{3} x^{9} + 42525 \, a b c^{7} d^{3} x^{7} + 59535 \, a b c^{5} d^{3} x^{5} + 33075 \, a b c^{3} d^{3} x^{3} - {\left (1225 \, b^{2} c^{8} d^{3} x^{8} + 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} + 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 630 \, {\left (1225 \, a b c^{8} d^{3} x^{8} + 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} + 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{31255875 \, c^{3}} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/31255875*(42875*(81*a^2 + 2*b^2)*c^9*d^3*x^9 + 1125*(11907*a^2 + 374*b^2 )*c^7*d^3*x^7 + 189*(99225*a^2 + 4198*b^2)*c^5*d^3*x^5 + 105*(99225*a^2 + 5258*b^2)*c^3*d^3*x^3 - 3312540*b^2*c*d^3*x + 99225*(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)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 630*(11025*a*b*c^9*d^3*x^9 + 42525*a*b*c^7*d^3*x^7 + 59535*a*b*c^5*d^3*x^5 + 33075*a*b*c^3*d^3*x^3 - (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)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 630*(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)*sqrt(c^2*x^2 + 1))/c^3
Time = 1.70 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.64 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(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 {asinh}{\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 {asinh}{\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 {asinh}{\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 {asinh}{\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 {asinh}^{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 {asinh}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asinh}^{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 {asinh}{\left (c x \right )}}{3969} + \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asinh}^{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 {asinh}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asinh}^{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 {asinh}{\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 {asinh}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**6*d**3*x**9/9 + 3*a**2*c**4*d**3*x**7/7 + 3*a**2*c**2*d **3*x**5/5 + a**2*d**3*x**3/3 + 2*a*b*c**6*d**3*x**9*asinh(c*x)/9 - 2*a*b* c**5*d**3*x**8*sqrt(c**2*x**2 + 1)/81 + 6*a*b*c**4*d**3*x**7*asinh(c*x)/7 - 374*a*b*c**3*d**3*x**6*sqrt(c**2*x**2 + 1)/3969 + 6*a*b*c**2*d**3*x**5*a sinh(c*x)/5 - 4198*a*b*c*d**3*x**4*sqrt(c**2*x**2 + 1)/33075 + 2*a*b*d**3* x**3*asinh(c*x)/3 - 5258*a*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(99225*c) + 105 16*a*b*d**3*sqrt(c**2*x**2 + 1)/(99225*c**3) + b**2*c**6*d**3*x**9*asinh(c *x)**2/9 + 2*b**2*c**6*d**3*x**9/729 - 2*b**2*c**5*d**3*x**8*sqrt(c**2*x** 2 + 1)*asinh(c*x)/81 + 3*b**2*c**4*d**3*x**7*asinh(c*x)**2/7 + 374*b**2*c* *4*d**3*x**7/27783 - 374*b**2*c**3*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x )/3969 + 3*b**2*c**2*d**3*x**5*asinh(c*x)**2/5 + 4198*b**2*c**2*d**3*x**5/ 165375 - 4198*b**2*c*d**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/33075 + b**2 *d**3*x**3*asinh(c*x)**2/3 + 5258*b**2*d**3*x**3/297675 - 5258*b**2*d**3*x **2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(99225*c) - 10516*b**2*d**3*x/(99225*c* *2) + 10516*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(99225*c**3), Ne(c, 0 )), (a**2*d**3*x**3/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (340) = 680\).
Time = 0.07 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.41 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/9*b^2*c^6*d^3*x^9*arcsinh(c*x)^2 + 1/9*a^2*c^6*d^3*x^9 + 3/7*b^2*c^4*d^3 *x^7*arcsinh(c*x)^2 + 3/7*a^2*c^4*d^3*x^7 + 3/5*b^2*c^2*d^3*x^5*arcsinh(c* x)^2 + 2/2835*(315*x^9*arcsinh(c*x) - (35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*s qrt(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^6*d^3 - 2/893025*(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*arcsinh(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^6*d^3 + 3/5*a^2*c^2*d^3*x^5 + 6/245*(35*x^7*a rcsinh(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^4*d^3 - 2/8575*(105*(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*arcsinh(c*x) - (7 5*c^6*x^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*c^4*d^3 + 1/3*b^2 *d^3*x^3*arcsinh(c*x)^2 + 2/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1) *x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c ^2*d^3 - 2/375*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/ c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d^3 + 1/3*a^2*d^3*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(s qrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d^3 - 2/27*(3*...
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
Output:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
\[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{3} \left (22050 \mathit {asinh} \left (c x \right ) a b \,c^{9} x^{9}+85050 \mathit {asinh} \left (c x \right ) a b \,c^{7} x^{7}+119070 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}+66150 \mathit {asinh} \left (c x \right ) a b \,c^{3} x^{3}-2450 \sqrt {c^{2} x^{2}+1}\, a b \,c^{8} x^{8}-9350 \sqrt {c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-12594 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}-5258 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}+10516 \sqrt {c^{2} x^{2}+1}\, a b +99225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{8}d x \right ) b^{2} c^{9}+297675 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+297675 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+99225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+11025 a^{2} c^{9} x^{9}+42525 a^{2} c^{7} x^{7}+59535 a^{2} c^{5} x^{5}+33075 a^{2} c^{3} x^{3}\right )}{99225 c^{3}} \] Input:
int(x^2*(c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
Output:
(d**3*(22050*asinh(c*x)*a*b*c**9*x**9 + 85050*asinh(c*x)*a*b*c**7*x**7 + 1 19070*asinh(c*x)*a*b*c**5*x**5 + 66150*asinh(c*x)*a*b*c**3*x**3 - 2450*sqr t(c**2*x**2 + 1)*a*b*c**8*x**8 - 9350*sqrt(c**2*x**2 + 1)*a*b*c**6*x**6 - 12594*sqrt(c**2*x**2 + 1)*a*b*c**4*x**4 - 5258*sqrt(c**2*x**2 + 1)*a*b*c** 2*x**2 + 10516*sqrt(c**2*x**2 + 1)*a*b + 99225*int(asinh(c*x)**2*x**8,x)*b **2*c**9 + 297675*int(asinh(c*x)**2*x**6,x)*b**2*c**7 + 297675*int(asinh(c *x)**2*x**4,x)*b**2*c**5 + 99225*int(asinh(c*x)**2*x**2,x)*b**2*c**3 + 110 25*a**2*c**9*x**9 + 42525*a**2*c**7*x**7 + 59535*a**2*c**5*x**5 + 33075*a* *2*c**3*x**3))/(99225*c**3)