Integrand size = 28, antiderivative size = 540 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {160 b^2 d^2 \sqrt {d+c^2 d x^2}}{3969 c^4}-\frac {80 b^2 d \left (d+c^2 d x^2\right )^{3/2}}{11907 c^4}-\frac {4 b^2 \left (d+c^2 d x^2\right )^{5/2}}{1323 c^4}-\frac {50 b^2 \left (d+c^2 d x^2\right )^{7/2}}{27783 c^4 d}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{9/2}}{729 c^4 d^2}+\frac {4 b d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{189 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{21 \sqrt {1+c^2 x^2}}-\frac {38 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{441 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{81 \sqrt {1+c^2 x^2}}-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{63 c^4}+\frac {d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \] Output:
-160/3969*b^2*d^2*(c^2*d*x^2+d)^(1/2)/c^4-80/11907*b^2*d*(c^2*d*x^2+d)^(3/ 2)/c^4-4/1323*b^2*(c^2*d*x^2+d)^(5/2)/c^4-50/27783*b^2*(c^2*d*x^2+d)^(7/2) /c^4/d+2/729*b^2*(c^2*d*x^2+d)^(9/2)/c^4/d^2+4/63*b*d^2*x*(c^2*d*x^2+d)^(1 /2)*(a+b*arcsinh(c*x))/c^3/(c^2*x^2+1)^(1/2)-2/189*b*d^2*x^3*(c^2*d*x^2+d) ^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*x^2+1)^(1/2)-2/21*b*c*d^2*x^5*(c^2*d*x^2+ d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-38/441*b*c^3*d^2*x^7*(c^2*d* x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-2/81*b*c^5*d^2*x^9*(c^2* d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-2/63*d^2*(c^2*d*x^2+d) ^(1/2)*(a+b*arcsinh(c*x))^2/c^4+1/63*d^2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcs inh(c*x))^2/c^2+1/21*d^2*x^4*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2+5/63 *d*x^4*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/9*x^4*(c^2*d*x^2+d)^(5/2 )*(a+b*arcsinh(c*x))^2
Time = 0.44 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.51 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \left (3969 a^2 \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right )-126 a b c x \sqrt {1+c^2 x^2} \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \left (-6140-7039 c^2 x^2+106 c^4 x^4+2152 c^6 x^6+1490 c^8 x^8+343 c^{10} x^{10}\right )-126 b \left (-63 a \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )\right ) \text {arcsinh}(c x)+3969 b^2 \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{250047 c^4 \left (1+c^2 x^2\right )} \] Input:
Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^2*Sqrt[d + c^2*d*x^2]*(3969*a^2*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - 126* a*b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(-6140 - 7039*c^2*x^2 + 106*c^4*x^4 + 2152*c^6*x^6 + 1490*c^8*x^8 + 343*c^10*x^10) - 126*b*(-63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x ^2) + b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x ^6 + 49*c^8*x^8))*ArcSinh[c*x] + 3969*b^2*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) *ArcSinh[c*x]^2))/(250047*c^4*(1 + c^2*x^2))
Time = 4.89 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.43, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6223, 6218, 27, 1578, 1195, 2009, 6223, 6218, 27, 354, 86, 2009, 6221, 6191, 243, 53, 2009, 6227, 6191, 243, 53, 2009, 6213, 2009}
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.
| \(\displaystyle \int x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{9 \sqrt {c^2 x^2+1}}+\frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{315 \sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{9 \sqrt {c^2 x^2+1}}+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{315} b c \int \frac {x^5 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{9 \sqrt {c^2 x^2+1}}+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{630} b c \int \frac {x^4 \left (35 c^4 x^4+90 c^2 x^2+63\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{9 \sqrt {c^2 x^2+1}}+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{630} b c \int \left (\frac {35 \left (c^2 x^2+1\right )^{7/2}}{c^4}-\frac {50 \left (c^2 x^2+1\right )^{5/2}}{c^4}+\frac {3 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {4 \sqrt {c^2 x^2+1}}{c^4}+\frac {8}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{9 \sqrt {c^2 x^2+1}}+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {c^2 d x^2+d} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{7 \sqrt {c^2 x^2+1}}+\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^5 \left (5 c^2 x^2+7\right )}{35 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{35} b c \int \frac {x^5 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \frac {x^4 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \left (\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^4}-\frac {8 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {\sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int x^4 (a+b \text {arcsinh}(c x))dx}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c^2 x^2+1}}dx\right )}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \left (\frac {\left (c^2 x^2+1\right )^{3/2}}{c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {1}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arcsinh}(c x))dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\right )+\frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{9} c^4 x^9 (a+b \text {arcsinh}(c x))+\frac {2}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{630} b c \left (\frac {70 \left (c^2 x^2+1\right )^{9/2}}{9 c^6}-\frac {100 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{9 \sqrt {c^2 x^2+1}}+\frac {5}{9} d \left (\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}+\frac {3}{7} d \left (\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}\right )\right )\) |
Input:
Int[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(x^4*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/9 - (2*b*c*d^2*Sqrt[d + c^2*d*x^2]*(-1/630*(b*c*((16*Sqrt[1 + c^2*x^2])/c^6 + (8*(1 + c^2*x^2)^(3 /2))/(3*c^6) + (6*(1 + c^2*x^2)^(5/2))/(5*c^6) - (100*(1 + c^2*x^2)^(7/2)) /(7*c^6) + (70*(1 + c^2*x^2)^(9/2))/(9*c^6))) + (x^5*(a + b*ArcSinh[c*x])) /5 + (2*c^2*x^7*(a + b*ArcSinh[c*x]))/7 + (c^4*x^9*(a + b*ArcSinh[c*x]))/9 ))/(9*Sqrt[1 + c^2*x^2]) + (5*d*((x^4*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh [c*x])^2)/7 - (2*b*c*d*Sqrt[d + c^2*d*x^2]*(-1/70*(b*c*((4*Sqrt[1 + c^2*x^ 2])/c^6 + (2*(1 + c^2*x^2)^(3/2))/(3*c^6) - (16*(1 + c^2*x^2)^(5/2))/(5*c^ 6) + (10*(1 + c^2*x^2)^(7/2))/(7*c^6))) + (x^5*(a + b*ArcSinh[c*x]))/5 + ( c^2*x^7*(a + b*ArcSinh[c*x]))/7))/(7*Sqrt[1 + c^2*x^2]) + (3*d*((x^4*Sqrt[ d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*Sqrt[d + c^2*d*x^2]*(-1/ 10*(b*c*((2*Sqrt[1 + c^2*x^2])/c^6 - (4*(1 + c^2*x^2)^(3/2))/(3*c^6) + (2* (1 + c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSinh[c*x]))/5))/(5*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2]*((x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[ c*x])^2)/(3*c^2) - (2*b*(-1/6*(b*c*((-2*Sqrt[1 + c^2*x^2])/c^4 + (2*(1 + c ^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSinh[c*x]))/3))/(3*c) - (2*((Sqr t[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt[1 + c^2*x ^2])/c + b*x*ArcSinh[c*x]))/c))/(3*c^2)))/(5*Sqrt[1 + c^2*x^2])))/7))/9
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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
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 = 1.34 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.01
| method | result | size |
| orering | \(\frac {\left (74431 c^{12} x^{12}+287825 c^{10} x^{10}+383719 c^{8} x^{8}+112267 c^{6} x^{6}-485758 c^{4} x^{4}-324980 c^{2} x^{2}-73680\right ) \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{250047 c^{6} \left (c^{2} x^{2}+1\right )^{3} x^{2}}-\frac {4 \left (686 c^{10} x^{10}+2503 c^{8} x^{8}+2822 c^{6} x^{6}-511 c^{4} x^{4}-8786 c^{2} x^{2}-3070\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+5 x^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{83349 x^{4} c^{6} \left (c^{2} x^{2}+1\right )^{2}}+\frac {\left (343 c^{8} x^{8}+1147 c^{6} x^{6}+1005 c^{4} x^{4}-899 c^{2} x^{2}-6140\right ) \left (6 x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+35 x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {12 x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+15 x^{5} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {20 x^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{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 )}{250047 c^{6} \left (c^{2} x^{2}+1\right ) x^{3}}\) | \(547\) |
| default | \(\text {Expression too large to display}\) | \(2014\) |
| parts | \(\text {Expression too large to display}\) | \(2014\) |
Input:
int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/250047*(74431*c^12*x^12+287825*c^10*x^10+383719*c^8*x^8+112267*c^6*x^6-4 85758*c^4*x^4-324980*c^2*x^2-73680)/c^6/(c^2*x^2+1)^3/x^2*(c^2*d*x^2+d)^(5 /2)*(a+b*arcsinh(x*c))^2-4/83349*(686*c^10*x^10+2503*c^8*x^8+2822*c^6*x^6- 511*c^4*x^4-8786*c^2*x^2-3070)/x^4/c^6/(c^2*x^2+1)^2*(3*x^2*(c^2*d*x^2+d)^ (5/2)*(a+b*arcsinh(x*c))^2+5*x^4*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))^2* c^2*d+2*x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))*b*c/(c^2*x^2+1)^(1/2))+ 1/250047*(343*c^8*x^8+1147*c^6*x^6+1005*c^4*x^4-899*c^2*x^2-6140)/c^6/(c^2 *x^2+1)/x^3*(6*x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2+35*x^3*(c^2*d*x^ 2+d)^(3/2)*(a+b*arcsinh(x*c))^2*c^2*d+12*x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcs inh(x*c))*b*c/(c^2*x^2+1)^(1/2)+15*x^5*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x* c))^2*c^4*d^2+20*x^4*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))*c^3*d*b/(c^2*x ^2+1)^(1/2)+2*x^3*(c^2*d*x^2+d)^(5/2)*b^2*c^2/(c^2*x^2+1)-2*x^4*(c^2*d*x^2 +d)^(5/2)*(a+b*arcsinh(x*c))*b*c^3/(c^2*x^2+1)^(3/2))
Time = 0.13 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} + 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} + 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} - 2 \, b^{2} d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 126 \, {\left (441 \, a b c^{10} d^{2} x^{10} + 1638 \, a b c^{8} d^{2} x^{8} + 2142 \, a b c^{6} d^{2} x^{6} + 1008 \, a b c^{4} d^{2} x^{4} - 63 \, a b c^{2} d^{2} x^{2} - 126 \, a b d^{2} - {\left (49 \, b^{2} c^{9} d^{2} x^{9} + 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} + 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (343 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} + 2 \, {\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} + 4 \, {\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} - 2 \, {\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2} - 126 \, {\left (49 \, a b c^{9} d^{2} x^{9} + 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} + 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} + c^{4}\right )}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="frica s")
Output:
1/250047*(3969*(7*b^2*c^10*d^2*x^10 + 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2* x^6 + 16*b^2*c^4*d^2*x^4 - b^2*c^2*d^2*x^2 - 2*b^2*d^2)*sqrt(c^2*d*x^2 + d )*log(c*x + sqrt(c^2*x^2 + 1))^2 + 126*(441*a*b*c^10*d^2*x^10 + 1638*a*b*c ^8*d^2*x^8 + 2142*a*b*c^6*d^2*x^6 + 1008*a*b*c^4*d^2*x^4 - 63*a*b*c^2*d^2* x^2 - 126*a*b*d^2 - (49*b^2*c^9*d^2*x^9 + 171*b^2*c^7*d^2*x^7 + 189*b^2*c^ 5*d^2*x^5 + 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt( c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (343*(81*a^2 + 2*b^2)*c^10*d ^2*x^10 + 2*(51597*a^2 + 1490*b^2)*c^8*d^2*x^8 + 2*(67473*a^2 + 2152*b^2)* c^6*d^2*x^6 + 4*(15876*a^2 + 53*b^2)*c^4*d^2*x^4 - (3969*a^2 + 14078*b^2)* c^2*d^2*x^2 - 2*(3969*a^2 + 6140*b^2)*d^2 - 126*(49*a*b*c^9*d^2*x^9 + 171* a*b*c^7*d^2*x^7 + 189*a*b*c^5*d^2*x^5 + 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2 *x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**3*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.72 \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {c^{2} x^{2} + 1} c^{6} d^{\frac {5}{2}} x^{8} + 1147 \, \sqrt {c^{2} x^{2} + 1} c^{4} d^{\frac {5}{2}} x^{6} + 1005 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} - 899 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} - \frac {6140 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {5}{2}}}{c^{2}}}{c^{2}} - \frac {63 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} + 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} + 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} + 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} + 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} a b}{3969 \, c^{3}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxim a")
Output:
1/63*(7*(c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(7/2)/(c^4*d ))*b^2*arcsinh(c*x)^2 + 2/63*(7*(c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2 *d*x^2 + d)^(7/2)/(c^4*d))*a*b*arcsinh(c*x) + 1/63*(7*(c^2*d*x^2 + d)^(7/2 )*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(7/2)/(c^4*d))*a^2 + 2/250047*b^2*((343* sqrt(c^2*x^2 + 1)*c^6*d^(5/2)*x^8 + 1147*sqrt(c^2*x^2 + 1)*c^4*d^(5/2)*x^6 + 1005*sqrt(c^2*x^2 + 1)*c^2*d^(5/2)*x^4 - 899*sqrt(c^2*x^2 + 1)*d^(5/2)* x^2 - 6140*sqrt(c^2*x^2 + 1)*d^(5/2)/c^2)/c^2 - 63*(49*c^8*d^(5/2)*x^9 + 1 71*c^6*d^(5/2)*x^7 + 189*c^4*d^(5/2)*x^5 + 21*c^2*d^(5/2)*x^3 - 126*d^(5/2 )*x)*arcsinh(c*x)/c^3) - 2/3969*(49*c^8*d^(5/2)*x^9 + 171*c^6*d^(5/2)*x^7 + 189*c^4*d^(5/2)*x^5 + 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*a*b/c^3
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2),x)
Output:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2), x)
\[ \int x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (7 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{8} x^{8}+19 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{6} x^{6}+15 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+\sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a^{2}+126 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{7}d x \right ) a b \,c^{8}+252 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) a b \,c^{6}+126 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}+126 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}\right )}{63 c^{4}} \] Input:
int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2,x)
Output:
(sqrt(d)*d**2*(7*sqrt(c**2*x**2 + 1)*a**2*c**8*x**8 + 19*sqrt(c**2*x**2 + 1)*a**2*c**6*x**6 + 15*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 + sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a**2 + 126*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**7,x)*a*b*c**8 + 252*int(sqrt(c**2*x**2 + 1)*asinh(c*x) *x**5,x)*a*b*c**6 + 126*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*a*b*c** 4 + 63*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**7,x)*b**2*c**8 + 126*int(s qrt(c**2*x**2 + 1)*asinh(c*x)**2*x**5,x)*b**2*c**6 + 63*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**3,x)*b**2*c**4))/(63*c**4)