Integrand size = 23, antiderivative size = 305 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{128 d}+\frac {35 b^2 e (a+b \text {arcsinh}(c+d x))^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}}{2 d}-\frac {105 b^{7/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 b^{7/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{1024 d} \]
35/64*b^2*e*(a+b*arcsinh(d*x+c))^(3/2)/d+35/32*b^2*e*(d*x+c)^2*(a+b*arcsin h(d*x+c))^(3/2)/d+1/4*e*(a+b*arcsinh(d*x+c))^(7/2)/d+1/2*e*(d*x+c)^2*(a+b* arcsinh(d*x+c))^(7/2)/d-105/2048*b^(7/2)*e*exp(2*a/b)*erf(2^(1/2)*(a+b*arc sinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+105/2048*b^(7/2)*e*erfi(2^( 1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-7/8 *b*e*(d*x+c)*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d-105/128*b^3* e*(d*x+c)*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d
Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.41 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {e e^{-\frac {2 a}{b}} \left (b^4 \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+b^4 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {9}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{64 \sqrt {2} d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
(e*(b^4*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (-2*(a + b*ArcSinh[ c + d*x]))/b] + b^4*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (2 *(a + b*ArcSinh[c + d*x]))/b]))/(64*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*ArcSi nh[c + d*x]])
Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {6274, 27, 6192, 6227, 6192, 6198, 6227, 6195, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198}
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.
| \(\displaystyle \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \int (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \int \frac {(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \int \frac {(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{4} b \int \frac {c+d x}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{4} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{8} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{8} \int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{8} i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{8} i \left (\frac {1}{2} i \int \frac {e^{\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{8} i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-i \int e^{\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{8} i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{2} \int \frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{8} i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{4} b \left (-\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{4} b \left (-\frac {1}{8} i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{3 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^{7/2}}{7 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}\right )\right )}{d}\) |
(e*(((c + d*x)^2*(a + b*ArcSinh[c + d*x])^(7/2))/2 - (7*b*(((c + d*x)*Sqrt [1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/2 - (a + b*ArcSinh[c + d *x])^(7/2)/(7*b) - (5*b*(((c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/2 - (3*b*(((c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/2 - ( a + b*ArcSinh[c + d*x])^(3/2)/(3*b) - (I/8)*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqr t[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - ((I/2)*Sqrt[ b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/E^((2* a)/b))))/4))/4))/4))/d
3.3.1.3.1 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[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 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_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -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_) + (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]
\[\int \left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]
Exception generated. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]