Integrand size = 28, antiderivative size = 28 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \text {Int}\left (x^m \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2,x\right )}{\sqrt {1+c^2 x^2}} \] Output:
d^2*(c^2*d*x^2+d)^(1/2)*Defer(Int)(x^m*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x) )^2,x)/(c^2*x^2+1)^(1/2)
Not integrable
Time = 3.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx \] Input:
Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x]
Not integrable
Time = 4.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int x^m \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \int x^{m+1} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx}{(m+6) \sqrt {c^2 x^2+1}}+\frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^{m+2} \left (\frac {c^4 x^4}{m+6}+\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {c^2 x^2+1}}dx+\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+6) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \left (\frac {\int \frac {c^2 x^{m+2} \left (\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}+\frac {m+6}{m+2}\right )}{\sqrt {c^2 x^2+1}}dx}{c^2 (m+6)}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+6) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \left (\frac {\int \frac {x^{m+2} \left (\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}+\frac {m+6}{m+2}\right )}{\sqrt {c^2 x^2+1}}dx}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+6) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) \int \frac {x^{m+2}}{\sqrt {c^2 x^2+1}}dx}{(m+2) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+6) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \int x^m \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {c^2 d x^2+d} \int x^{m+1} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{(m+4) \sqrt {c^2 x^2+1}}+\frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^{m+2} \left (\frac {c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {c^2 x^2+1}}dx+\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5 d \left (-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \left (\frac {(3 m+10) \int \frac {x^{m+2}}{\sqrt {c^2 x^2+1}}dx}{(m+2) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )+\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int x^{m+1} (a+b \text {arcsinh}(c x))dx}{(m+2) \sqrt {c^2 x^2+1}}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c \int \frac {x^{m+2}}{\sqrt {c^2 x^2+1}}dx}{m+2}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6239 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{m+6}-\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {c^4 x^{m+6} (a+b \text {arcsinh}(c x))}{m+6}+\frac {2 c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}+\frac {c^2 \sqrt {c^2 x^2+1} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {c^2 x^2+1}}\) |
Input:
Int[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
$Aborted
Not integrable
Time = 2.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int x^{m} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}d x\]
Input:
int(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2,x)
Output:
int(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2,x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)*x^m, x)
Timed out. \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**m*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)
Output:
Timed out
Not integrable
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxim a")
Output:
integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^2*x^m, x)
Exception generated. \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*(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
Not integrable
Time = 3.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:
int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2),x)
Output:
int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2), x)
Not integrable
Time = 0.31 (sec) , antiderivative size = 255, normalized size of antiderivative = 9.11 \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\sqrt {d}\, d^{2} \left (2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) a b \,c^{4}+4 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b +\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{4}+2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, x^{4}d x \right ) a^{2} c^{4}+2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, x^{2}d x \right ) a^{2} c^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}d x \right ) a^{2}\right ) \] Input:
int(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2,x)
Output:
sqrt(d)*d**2*(2*int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*a*b*c**4 + 4*int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**2 + 2*int(x**m*s qrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b + int(x**m*sqrt(c**2*x**2 + 1)*asinh( c*x)**2*x**4,x)*b**2*c**4 + 2*int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x **2,x)*b**2*c**2 + int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x)**2,x)*b**2 + in t(x**m*sqrt(c**2*x**2 + 1)*x**4,x)*a**2*c**4 + 2*int(x**m*sqrt(c**2*x**2 + 1)*x**2,x)*a**2*c**2 + int(x**m*sqrt(c**2*x**2 + 1),x)*a**2)