Integrand size = 25, antiderivative size = 701 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}}{5 d}+\frac {15 b^{5/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {b^{5/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{240 d}-\frac {b^{5/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{1280 d}+\frac {3 b^{5/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{6400 d}-\frac {15 b^{5/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {b^{5/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{240 d}+\frac {b^{5/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{1280 d}-\frac {3 b^{5/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{6400 d} \]
1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^(5/2)/d+3/32000*b^(5/2)*e^4*exp(5*a /b)*erf(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/d-3/3 2000*b^(5/2)*e^4*erfi(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)* Pi^(1/2)/d/exp(5*a/b)-5/2304*b^(5/2)*e^4*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsi nh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/d+5/2304*b^(5/2)*e^4*erfi(3^(1/ 2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/d/exp(3*a/b)+15/12 8*b^(5/2)*e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d- 15/128*b^(5/2)*e^4*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp (a/b)-4/15*b*e^4*(a+b*arcsinh(d*x+c))^(3/2)*(1+(d*x+c)^2)^(1/2)/d+2/15*b*e ^4*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(3/2)*(1+(d*x+c)^2)^(1/2)/d-1/10*b*e^4*( d*x+c)^4*(a+b*arcsinh(d*x+c))^(3/2)*(1+(d*x+c)^2)^(1/2)/d+2/5*b^2*e^4*(d*x +c)*(a+b*arcsinh(d*x+c))^(1/2)/d-1/15*b^2*e^4*(d*x+c)^3*(a+b*arcsinh(d*x+c ))^(1/2)/d+3/100*b^2*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^(1/2)/d
Time = 0.37 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.46 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {b^3 e^4 e^{-\frac {5 a}{b}} \left (33750 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+27 \sqrt {5} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )-625 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+33750 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-625 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+27 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{540000 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
-1/540000*(b^3*e^4*(33750*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7 /2, a/b + ArcSinh[c + d*x]] + 27*Sqrt[5]*Sqrt[-((a + b*ArcSinh[c + d*x])/b )]*Gamma[7/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 625*Sqrt[3]*E^((2*a)/b)*S qrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (-3*(a + b*ArcSinh[c + d*x]) )/b] + 33750*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, -( (a + b*ArcSinh[c + d*x])/b)] - 625*Sqrt[3]*E^((8*a)/b)*Sqrt[a/b + ArcSinh[ c + d*x]]*Gamma[7/2, (3*(a + b*ArcSinh[c + d*x]))/b] + 27*Sqrt[5]*E^((10*a )/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (5*(a + b*ArcSinh[c + d*x]))/ b]))/(d*E^((5*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])
Result contains complex when optimal does not.
Time = 4.06 (sec) , antiderivative size = 853, normalized size of antiderivative = 1.22, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {6274, 27, 6192, 6227, 6192, 6227, 6192, 6213, 6187, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634, 3793, 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.
| \(\displaystyle \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^4 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \int \frac {(c+d x)^5 (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \int (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {4}{5} \left (-\frac {1}{2} b \int (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \int \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} b \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} \int -\frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} \int -\frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {3}{10} b \left (\frac {1}{10} \int \frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{6} \int \frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} \int \frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{10} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-i \int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (-\frac {4}{5} \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \left (\frac {3 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {1}{2} b \left (\frac {1}{5} \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2} (c+d x)^4-\frac {4}{5} \left (\frac {1}{3} \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2} (c+d x)^2-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} \sqrt {a+b \text {arcsinh}(c+d x)} (c+d x)^3+\frac {1}{6} i \left (\frac {3}{8} i \sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} i \sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{8} i \sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} i \sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )-\frac {3}{10} b \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \left (\frac {5}{16} i \sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{32} i \sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} i \sqrt {b} e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{16} i \sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {5}{32} i \sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} i \sqrt {b} e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )\right )}{d}\) |
(e^4*(((c + d*x)^5*(a + b*ArcSinh[c + d*x])^(5/2))/5 - (b*(((c + d*x)^4*Sq rt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/5 - (4*(((c + d*x)^2*S qrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/3 - (2*(Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2) - (3*b*((c + d*x)*Sqrt[a + b*ArcSi nh[c + d*x]] - (I/2)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSin h[c + d*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b))))/2))/3 - (b*(((c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/3 + (I/6)*(((3*I)/8)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcS inh[c + d*x]]/Sqrt[b]] - (I/8)*Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3] *Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - (((3*I)/8)*Sqrt[b]*Sqrt[Pi]*Erfi [Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b) + ((I/8)*Sqrt[b]*Sqrt[Pi/3 ]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/E^((3*a)/b))))/2)) /5 - (3*b*(((c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/5 - (I/10)*(((5*I)/1 6)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - (( 5*I)/32)*Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + (I/32)*Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqr t[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - (((5*I)/16)*Sqrt[b]*Sqrt[Pi]*Erfi[Sq rt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b) + (((5*I)/32)*Sqrt[b]*Sqrt[Pi /3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/E^((3*a)/b) - (( I/32)*Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sq...
3.2.92.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
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_.))^(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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
\[\int \left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]