Integrand size = 26, antiderivative size = 303 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {1636 b^2 d^2 x}{11025 c^2}+\frac {818 b^2 d^2 x^3}{33075}+\frac {136 b^2 c^2 d^2 x^5}{6125}+\frac {2}{343} b^2 c^4 d^2 x^7+\frac {32 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{315 c^3}-\frac {16 b d^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{315 c}+\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{105 c^3}+\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{175 c^3}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{49 c^3}+\frac {8}{105} d^2 x^3 (a+b \text {arcsinh}(c x))^2+\frac {4}{35} d^2 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{7} d^2 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \]
-1636/11025*b^2*d^2*x/c^2+818/33075*b^2*d^2*x^3+136/6125*b^2*c^2*d^2*x^5+2 /343*b^2*c^4*d^2*x^7+8/105*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^3+ 2/175*b*d^2*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^3-2/49*b*d^2*(c^2*x^2+1 )^(7/2)*(a+b*arcsinh(c*x))/c^3+8/105*d^2*x^3*(a+b*arcsinh(c*x))^2+4/35*d^2 *x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/7*d^2*x^3*(c^2*x^2+1)^2*(a+b*arcsi nh(c*x))^2+32/315*b*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-16/315*b* d^2*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 1.13 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.75 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (11025 a^2 c^3 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right )-210 a b \sqrt {1+c^2 x^2} \left (-818+409 c^2 x^2+612 c^4 x^4+225 c^6 x^6\right )+2 b^2 c x \left (-85890+14315 c^2 x^2+12852 c^4 x^4+3375 c^6 x^6\right )-210 b \left (-105 a c^3 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (-818+409 c^2 x^2+612 c^4 x^4+225 c^6 x^6\right )\right ) \text {arcsinh}(c x)+11025 b^2 c^3 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{1157625 c^3} \]
(d^2*(11025*a^2*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) - 210*a*b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6) + 2*b^2*c*x*(-85 890 + 14315*c^2*x^2 + 12852*c^4*x^4 + 3375*c^6*x^6) - 210*b*(-105*a*c^3*x^ 3*(35 + 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c^3*x^3*(35 + 42*c ^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x]^2))/(1157625*c^3)
Time = 1.88 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {6223, 27, 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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2}{7} b c d^2 \int x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{7} d \int d x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c d^2 \int x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c d^2 \left (-b c \int -\frac {\left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2}{35 c^4}dx+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c d^2 \left (\frac {b \int \left (2-5 c^2 x^2\right ) \left (c^2 x^2+1\right )^2dx}{35 c^3}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{7} b c d^2 \left (\frac {b \int \left (-5 c^6 x^6-8 c^4 x^4-c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {4}{7} d^2 \left (-\frac {2}{5} b c \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{5} \int x^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \int x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \left (-b c \int -\frac {-3 c^4 x^4-c^2 x^2+2}{15 c^4}dx+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{5} b c \left (\frac {b \int \left (-3 c^4 x^4-c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{7} d^2 \left (\frac {1}{5} x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{3} b c \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )-\frac {2}{7} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4}-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4}+\frac {b \left (-\frac {5}{7} c^6 x^7-\frac {8 c^4 x^5}{5}-\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
(d^2*x^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/7 - (2*b*c*d^2*((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*d^2*((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*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)))/3))/5))/7
3.3.9.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((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 = 0.24 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{7} c^{4} x^{7}+\frac {2}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{105}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}-\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{105}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}-\frac {181456 c x}{1157625}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {202 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {2528 c x \left (c^{2} x^{2}+1\right )}{1157625}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{105}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}+\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{315}\right )}{c^{3}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {409 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}\right )}{c^{3}}\) | \(345\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{105}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}-\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{105}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}-\frac {181456 c x}{1157625}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {202 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {2528 c x \left (c^{2} x^{2}+1\right )}{1157625}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{105}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}+\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{315}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {409 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}\right )}{c^{3}}\) | \(346\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{105}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}-\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{105}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}-\frac {181456 c x}{1157625}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {202 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {2528 c x \left (c^{2} x^{2}+1\right )}{1157625}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{105}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}+\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{315}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {409 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}\right )}{c^{3}}\) | \(346\) |
d^2*a^2*(1/7*c^4*x^7+2/5*c^2*x^5+1/3*x^3)+d^2*b^2/c^3*(1/7*arcsinh(c*x)^2* c*x*(c^2*x^2+1)^3-8/105*arcsinh(c*x)^2*x*c-1/35*arcsinh(c*x)^2*c*x*(c^2*x^ 2+1)^2-4/105*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-2/49*arcsinh(c*x)*(c^2*x^2+1)^ (7/2)-181456/1157625*c*x+2/343*c*x*(c^2*x^2+1)^3+202/42875*c*x*(c^2*x^2+1) ^2-2528/1157625*c*x*(c^2*x^2+1)+16/105*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2/17 5*arcsinh(c*x)*(c^2*x^2+1)^(5/2)+8/315*arcsinh(c*x)*(c^2*x^2+1)^(3/2))+2*d ^2*a*b/c^3*(1/7*arcsinh(c*x)*c^7*x^7+2/5*arcsinh(c*x)*c^5*x^5+1/3*arcsinh( c*x)*c^3*x^3-1/49*c^6*x^6*(c^2*x^2+1)^(1/2)-68/1225*c^4*x^4*(c^2*x^2+1)^(1 /2)-409/11025*c^2*x^2*(c^2*x^2+1)^(1/2)+818/11025*(c^2*x^2+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.08 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {3375 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} x^{7} + 378 \, {\left (1225 \, a^{2} + 68 \, b^{2}\right )} c^{5} d^{2} x^{5} + 35 \, {\left (11025 \, a^{2} + 818 \, b^{2}\right )} c^{3} d^{2} x^{3} - 171780 \, b^{2} c d^{2} x + 11025 \, {\left (15 \, b^{2} c^{7} d^{2} x^{7} + 42 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (1575 \, a b c^{7} d^{2} x^{7} + 4410 \, a b c^{5} d^{2} x^{5} + 3675 \, a b c^{3} d^{2} x^{3} - {\left (225 \, b^{2} c^{6} d^{2} x^{6} + 612 \, b^{2} c^{4} d^{2} x^{4} + 409 \, b^{2} c^{2} d^{2} x^{2} - 818 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 210 \, {\left (225 \, a b c^{6} d^{2} x^{6} + 612 \, a b c^{4} d^{2} x^{4} + 409 \, a b c^{2} d^{2} x^{2} - 818 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{1157625 \, c^{3}} \]
1/1157625*(3375*(49*a^2 + 2*b^2)*c^7*d^2*x^7 + 378*(1225*a^2 + 68*b^2)*c^5 *d^2*x^5 + 35*(11025*a^2 + 818*b^2)*c^3*d^2*x^3 - 171780*b^2*c*d^2*x + 110 25*(15*b^2*c^7*d^2*x^7 + 42*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 210*(1575*a*b*c^7*d^2*x^7 + 4410*a*b*c^5*d^2*x^5 + 3675*a*b*c^3*d^2*x^3 - (225*b^2*c^6*d^2*x^6 + 612*b^2*c^4*d^2*x^4 + 409* b^2*c^2*d^2*x^2 - 818*b^2*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 210*(225*a*b*c^6*d^2*x^6 + 612*a*b*c^4*d^2*x^4 + 409*a*b*c^2*d^2*x^ 2 - 818*a*b*d^2)*sqrt(c^2*x^2 + 1))/c^3
Time = 0.98 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.59 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{7}}{7} + \frac {2 a^{2} c^{2} d^{2} x^{5}}{5} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {2 a b c^{4} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b c^{3} d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {4 a b c^{2} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {136 a b c d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + \frac {2 a b d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {818 a b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c} + \frac {1636 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c^{3}} + \frac {b^{2} c^{4} d^{2} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{4} d^{2} x^{7}}{343} - \frac {2 b^{2} c^{3} d^{2} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49} + \frac {2 b^{2} c^{2} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {136 b^{2} c^{2} d^{2} x^{5}}{6125} - \frac {136 b^{2} c d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1225} + \frac {b^{2} d^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {818 b^{2} d^{2} x^{3}}{33075} - \frac {818 b^{2} d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{11025 c} - \frac {1636 b^{2} d^{2} x}{11025 c^{2}} + \frac {1636 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**4*d**2*x**7/7 + 2*a**2*c**2*d**2*x**5/5 + a**2*d**2*x** 3/3 + 2*a*b*c**4*d**2*x**7*asinh(c*x)/7 - 2*a*b*c**3*d**2*x**6*sqrt(c**2*x **2 + 1)/49 + 4*a*b*c**2*d**2*x**5*asinh(c*x)/5 - 136*a*b*c*d**2*x**4*sqrt (c**2*x**2 + 1)/1225 + 2*a*b*d**2*x**3*asinh(c*x)/3 - 818*a*b*d**2*x**2*sq rt(c**2*x**2 + 1)/(11025*c) + 1636*a*b*d**2*sqrt(c**2*x**2 + 1)/(11025*c** 3) + b**2*c**4*d**2*x**7*asinh(c*x)**2/7 + 2*b**2*c**4*d**2*x**7/343 - 2*b **2*c**3*d**2*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/49 + 2*b**2*c**2*d**2*x* *5*asinh(c*x)**2/5 + 136*b**2*c**2*d**2*x**5/6125 - 136*b**2*c*d**2*x**4*s qrt(c**2*x**2 + 1)*asinh(c*x)/1225 + b**2*d**2*x**3*asinh(c*x)**2/3 + 818* b**2*d**2*x**3/33075 - 818*b**2*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/( 11025*c) - 1636*b**2*d**2*x/(11025*c**2) + 1636*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(11025*c**3), Ne(c, 0)), (a**2*d**2*x**3/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (269) = 538\).
Time = 0.22 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.04 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{7} \, b^{2} c^{4} d^{2} x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} c^{4} d^{2} x^{7} + \frac {2}{5} \, b^{2} c^{2} d^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{2} + \frac {1}{3} \, b^{2} d^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {4}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d^{2} - \frac {4}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{2} - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d^{2} \]
1/7*b^2*c^4*d^2*x^7*arcsinh(c*x)^2 + 1/7*a^2*c^4*d^2*x^7 + 2/5*b^2*c^2*d^2 *x^5*arcsinh(c*x)^2 + 2/5*a^2*c^2*d^2*x^5 + 2/245*(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^4*d^2 - 2/25725*(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 - 12 6*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*c^4*d^2 + 1/3*b^2*d^2*x^3*arcsi nh(c*x)^2 + 4/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*s qrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^2*d^2 - 4/112 5*(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^2 + 1/3*a^2*d^2*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d^2 - 2/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2 )*b^2*d^2
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]