Integrand size = 26, antiderivative size = 386 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4208 b^2 d^2 x}{99225 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}+\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}+\frac {2}{729} b^2 c^4 d^2 x^9-\frac {128 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{4725 c^3}-\frac {16 b d^2 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{1575 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{189 c^5}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{315 c^5}+\frac {20 b d^2 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^5}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^5}+\frac {8}{315} d^2 x^5 (a+b \text {arcsinh}(c x))^2+\frac {4}{63} d^2 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^2 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \] Output:
4208/99225*b^2*d^2*x/c^4-2104/297675*b^2*d^2*x^3/c^2+526/165375*b^2*d^2*x^ 5+212/27783*b^2*c^2*d^2*x^7+2/729*b^2*c^4*d^2*x^9-128/4725*b*d^2*(c^2*x^2+ 1)^(1/2)*(a+b*arcsinh(c*x))/c^5+64/4725*b*d^2*x^2*(c^2*x^2+1)^(1/2)*(a+b*a rcsinh(c*x))/c^3-16/1575*b*d^2*x^4*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c- 8/189*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^5+2/315*b*d^2*(c^2*x^2+ 1)^(5/2)*(a+b*arcsinh(c*x))/c^5+20/441*b*d^2*(c^2*x^2+1)^(7/2)*(a+b*arcsin h(c*x))/c^5-2/81*b*d^2*(c^2*x^2+1)^(9/2)*(a+b*arcsinh(c*x))/c^5+8/315*d^2* x^5*(a+b*arcsinh(c*x))^2+4/63*d^2*x^5*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/9 *d^2*x^5*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2
Time = 0.24 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.65 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (99225 a^2 c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )-630 a b \sqrt {1+c^2 x^2} \left (2104-1052 c^2 x^2+789 c^4 x^4+2650 c^6 x^6+1225 c^8 x^8\right )+2 b^2 c x \left (662760-110460 c^2 x^2+49707 c^4 x^4+119250 c^6 x^6+42875 c^8 x^8\right )-630 b \left (-315 a c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (2104-1052 c^2 x^2+789 c^4 x^4+2650 c^6 x^6+1225 c^8 x^8\right )\right ) \text {arcsinh}(c x)+99225 b^2 c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{31255875 c^5} \] Input:
Integrate[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*(99225*a^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) - 630*a*b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8) + 2*b^2*c*x*(662760 - 110460*c^2*x^2 + 49707*c^4*x^4 + 119250*c^6*x^6 + 4 2875*c^8*x^8) - 630*b*(-315*a*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) + b*S qrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225* c^8*x^8))*ArcSinh[c*x] + 99225*b^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4)* ArcSinh[c*x]^2))/(31255875*c^5)
Time = 3.40 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {6223, 27, 6219, 27, 1467, 2009, 6223, 6191, 6219, 27, 2009, 6227, 15, 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.
| \(\displaystyle \int x^4 \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2}{9} b c d^2 \int x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{9} d \int d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{9} b c d^2 \int x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{9} d^2 \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^2 \left (-b c \int \frac {\left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )}{315 c^6}dx+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^2 \left (-\frac {b \int \left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c d^2 \left (-\frac {b \int \left (35 c^8 x^8+50 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {4}{9} d^2 \left (-\frac {2}{7} b c \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{7} \int x^4 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \left (-b c \int \frac {15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8}{105 c^6}dx+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \left (-\frac {b \int \left (15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{105 c^5}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}-\frac {b \int x^4dx}{5 c}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} d^2 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{9} d^2 \left (\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {4 \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )-\frac {2}{9} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
Input:
Int[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/9 - (2*b*c*d^2*(-1/315*(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))/ c^5 + ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^6) - (2*(1 + c^2*x^2 )^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^6) + ((1 + c^2*x^2)^(9/2)*(a + b*ArcSin h[c*x]))/(9*c^6)))/9 + (4*d^2*((x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/ 7 - (2*b*c*(-1/105*(b*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (15*c^6*x^7)/ 7))/c^5 + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^6) - (2*(1 + c^2 *x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^6) + ((1 + c^2*x^2)^(7/2)*(a + b*Ar cSinh[c*x]))/(7*c^6)))/7 + (2*((x^5*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*(-1 /25*(b*x^5)/c + (x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) - (4* (-1/9*(b*x^3)/c + (x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[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)))/( 5*c^2)))/5))/7))/9
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_.) + 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 = 2.59 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.11
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{9} c^{4} x^{9}+\frac {2}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {b^{2} d^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{9}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{21}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{81}+\frac {82 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{3969}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {1493104 x c}{31255875}-\frac {836 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}-\frac {33862 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {47248 x c \left (c^{2} x^{2}+1\right )}{31255875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )}{c^{5}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{5}}\) | \(427\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b^{2} d^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{9}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{21}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{81}+\frac {82 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{3969}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {1493104 x c}{31255875}-\frac {836 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}-\frac {33862 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {47248 x c \left (c^{2} x^{2}+1\right )}{31255875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{5}}\) | \(428\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b^{2} d^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{9}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{21}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{315}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{105}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{81}+\frac {82 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{3969}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {1493104 x c}{31255875}-\frac {836 x c \left (c^{2} x^{2}+1\right )^{3}}{250047}-\frac {33862 x c \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {47248 x c \left (c^{2} x^{2}+1\right )}{31255875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{315}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{9}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {263 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}-\frac {106 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{81}\right )}{c^{5}}\) | \(428\) |
| orering | \(\frac {\left (9303875 c^{12} x^{12}+34087625 c^{10} x^{10}+40400953 c^{8} x^{8}+8418363 c^{6} x^{6}+38661000 c^{4} x^{4}+46835040 c^{2} x^{2}+15906240\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{31255875 x \,c^{6} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (1029000 c^{10} x^{10}+3352375 c^{8} x^{8}+2782890 c^{6} x^{6}-1342089 c^{4} x^{4}+9389100 c^{2} x^{2}+5964840\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+4 x^{5} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{31255875 x^{4} c^{6} \left (c^{2} x^{2}+1\right )^{2}}+\frac {\left (42875 c^{8} x^{8}+119250 c^{6} x^{6}+49707 c^{4} x^{4}-110460 c^{2} x^{2}+662760\right ) \left (12 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+36 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {16 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+8 x^{6} c^{4} d^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {16 x^{5} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{5} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{31255875 x^{3} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(531\) |
Input:
int(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
d^2*a^2*(1/9*c^4*x^9+2/7*c^2*x^7+1/5*x^5)+b^2*d^2/c^5*(1/9*x^3*c^3*(c^2*x^ 2+1)^3*arcsinh(x*c)^2-1/21*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^3+8/315*arcsinh( x*c)^2*x*c+1/105*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^2+4/315*arcsinh(x*c)^2*x*c *(c^2*x^2+1)-2/81*arcsinh(x*c)*x^2*c^2*(c^2*x^2+1)^(7/2)+82/3969*arcsinh(x *c)*(c^2*x^2+1)^(7/2)+2/729*x*c*(c^2*x^2+1)^4+1493104/31255875*x*c-836/250 047*x*c*(c^2*x^2+1)^3-33862/10418625*x*c*(c^2*x^2+1)^2-47248/31255875*x*c* (c^2*x^2+1)-16/315*arcsinh(x*c)*(c^2*x^2+1)^(1/2)-2/525*arcsinh(x*c)*(c^2* x^2+1)^(5/2)-8/945*arcsinh(x*c)*(c^2*x^2+1)^(3/2))+2*d^2*a*b/c^5*(1/9*arcs inh(x*c)*x^9*c^9+2/7*arcsinh(x*c)*c^7*x^7+1/5*arcsinh(x*c)*x^5*c^5-263/330 75*x^4*c^4*(c^2*x^2+1)^(1/2)+1052/99225*x^2*c^2*(c^2*x^2+1)^(1/2)-2104/992 25*(c^2*x^2+1)^(1/2)-106/3969*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 = 368, normalized size of antiderivative = 0.95 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {42875 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{2} x^{9} + 2250 \, {\left (3969 \, a^{2} + 106 \, b^{2}\right )} c^{7} d^{2} x^{7} + 189 \, {\left (33075 \, a^{2} + 526 \, b^{2}\right )} c^{5} d^{2} x^{5} - 220920 \, b^{2} c^{3} d^{2} x^{3} + 1325520 \, b^{2} c d^{2} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{2} x^{9} + 90 \, b^{2} c^{7} d^{2} x^{7} + 63 \, b^{2} c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 630 \, {\left (11025 \, a b c^{9} d^{2} x^{9} + 28350 \, a b c^{7} d^{2} x^{7} + 19845 \, a b c^{5} d^{2} x^{5} - {\left (1225 \, b^{2} c^{8} d^{2} x^{8} + 2650 \, b^{2} c^{6} d^{2} x^{6} + 789 \, b^{2} c^{4} d^{2} x^{4} - 1052 \, b^{2} c^{2} d^{2} x^{2} + 2104 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 630 \, {\left (1225 \, a b c^{8} d^{2} x^{8} + 2650 \, a b c^{6} d^{2} x^{6} + 789 \, a b c^{4} d^{2} x^{4} - 1052 \, a b c^{2} d^{2} x^{2} + 2104 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{31255875 \, c^{5}} \] Input:
integrate(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/31255875*(42875*(81*a^2 + 2*b^2)*c^9*d^2*x^9 + 2250*(3969*a^2 + 106*b^2) *c^7*d^2*x^7 + 189*(33075*a^2 + 526*b^2)*c^5*d^2*x^5 - 220920*b^2*c^3*d^2* x^3 + 1325520*b^2*c*d^2*x + 99225*(35*b^2*c^9*d^2*x^9 + 90*b^2*c^7*d^2*x^7 + 63*b^2*c^5*d^2*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 630*(11025*a*b*c^9 *d^2*x^9 + 28350*a*b*c^7*d^2*x^7 + 19845*a*b*c^5*d^2*x^5 - (1225*b^2*c^8*d ^2*x^8 + 2650*b^2*c^6*d^2*x^6 + 789*b^2*c^4*d^2*x^4 - 1052*b^2*c^2*d^2*x^2 + 2104*b^2*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 630*(12 25*a*b*c^8*d^2*x^8 + 2650*a*b*c^6*d^2*x^6 + 789*a*b*c^4*d^2*x^4 - 1052*a*b *c^2*d^2*x^2 + 2104*a*b*d^2)*sqrt(c^2*x^2 + 1))/c^5
Time = 1.75 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.46 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{9}}{9} + \frac {2 a^{2} c^{2} d^{2} x^{7}}{7} + \frac {a^{2} d^{2} x^{5}}{5} + \frac {2 a b c^{4} d^{2} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {2 a b c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {4 a b c^{2} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {212 a b c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {2 a b d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {526 a b d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075 c} + \frac {2104 a b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {4208 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{5}} + \frac {b^{2} c^{4} d^{2} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{4} d^{2} x^{9}}{729} - \frac {2 b^{2} c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{81} + \frac {2 b^{2} c^{2} d^{2} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {212 b^{2} c^{2} d^{2} x^{7}}{27783} - \frac {212 b^{2} c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3969} + \frac {b^{2} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {526 b^{2} d^{2} x^{5}}{165375} - \frac {526 b^{2} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{33075 c} - \frac {2104 b^{2} d^{2} x^{3}}{297675 c^{2}} + \frac {2104 b^{2} d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{3}} + \frac {4208 b^{2} d^{2} x}{99225 c^{4}} - \frac {4208 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(c**2*d*x**2+d)**2*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**4*d**2*x**9/9 + 2*a**2*c**2*d**2*x**7/7 + a**2*d**2*x** 5/5 + 2*a*b*c**4*d**2*x**9*asinh(c*x)/9 - 2*a*b*c**3*d**2*x**8*sqrt(c**2*x **2 + 1)/81 + 4*a*b*c**2*d**2*x**7*asinh(c*x)/7 - 212*a*b*c*d**2*x**6*sqrt (c**2*x**2 + 1)/3969 + 2*a*b*d**2*x**5*asinh(c*x)/5 - 526*a*b*d**2*x**4*sq rt(c**2*x**2 + 1)/(33075*c) + 2104*a*b*d**2*x**2*sqrt(c**2*x**2 + 1)/(9922 5*c**3) - 4208*a*b*d**2*sqrt(c**2*x**2 + 1)/(99225*c**5) + b**2*c**4*d**2* x**9*asinh(c*x)**2/9 + 2*b**2*c**4*d**2*x**9/729 - 2*b**2*c**3*d**2*x**8*s qrt(c**2*x**2 + 1)*asinh(c*x)/81 + 2*b**2*c**2*d**2*x**7*asinh(c*x)**2/7 + 212*b**2*c**2*d**2*x**7/27783 - 212*b**2*c*d**2*x**6*sqrt(c**2*x**2 + 1)* asinh(c*x)/3969 + b**2*d**2*x**5*asinh(c*x)**2/5 + 526*b**2*d**2*x**5/1653 75 - 526*b**2*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(33075*c) - 2104*b* *2*d**2*x**3/(297675*c**2) + 2104*b**2*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh (c*x)/(99225*c**3) + 4208*b**2*d**2*x/(99225*c**4) - 4208*b**2*d**2*sqrt(c **2*x**2 + 1)*asinh(c*x)/(99225*c**5), Ne(c, 0)), (a**2*d**2*x**5/5, True) )
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (342) = 684\).
Time = 0.06 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.97 \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/9*b^2*c^4*d^2*x^9*arcsinh(c*x)^2 + 1/9*a^2*c^4*d^2*x^9 + 2/7*b^2*c^2*d^2 *x^7*arcsinh(c*x)^2 + 2/7*a^2*c^2*d^2*x^7 + 1/5*b^2*d^2*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*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^2 - 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^4*d^2 + 1/5*a^2*d^2*x^5 + 4/245*(35*x^7*arcsinh(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^2 - 4/25725* (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) - (75*c^6*x ^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*c^2*d^2 + 2/75*(15*x^5*a rcsinh(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*d^2 - 2/1125*(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*d^2
Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(c^2*d*x^2+d)^2*(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. \[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:
int(x^4*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2,x)
Output:
int(x^4*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2, x)
\[ \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{2} \left (22050 \mathit {asinh} \left (c x \right ) a b \,c^{9} x^{9}+56700 \mathit {asinh} \left (c x \right ) a b \,c^{7} x^{7}+39690 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}-2450 \sqrt {c^{2} x^{2}+1}\, a b \,c^{8} x^{8}-5300 \sqrt {c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-1578 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}+2104 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-4208 \sqrt {c^{2} x^{2}+1}\, a b +99225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{8}d x \right ) b^{2} c^{9}+198450 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+99225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+11025 a^{2} c^{9} x^{9}+28350 a^{2} c^{7} x^{7}+19845 a^{2} c^{5} x^{5}\right )}{99225 c^{5}} \] Input:
int(x^4*(c^2*d*x^2+d)^2*(a+b*asinh(c*x))^2,x)
Output:
(d**2*(22050*asinh(c*x)*a*b*c**9*x**9 + 56700*asinh(c*x)*a*b*c**7*x**7 + 3 9690*asinh(c*x)*a*b*c**5*x**5 - 2450*sqrt(c**2*x**2 + 1)*a*b*c**8*x**8 - 5 300*sqrt(c**2*x**2 + 1)*a*b*c**6*x**6 - 1578*sqrt(c**2*x**2 + 1)*a*b*c**4* x**4 + 2104*sqrt(c**2*x**2 + 1)*a*b*c**2*x**2 - 4208*sqrt(c**2*x**2 + 1)*a *b + 99225*int(asinh(c*x)**2*x**8,x)*b**2*c**9 + 198450*int(asinh(c*x)**2* x**6,x)*b**2*c**7 + 99225*int(asinh(c*x)**2*x**4,x)*b**2*c**5 + 11025*a**2 *c**9*x**9 + 28350*a**2*c**7*x**7 + 19845*a**2*c**5*x**5))/(99225*c**5)