Integrand size = 25, antiderivative size = 410 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d} \]
-8/15*e^2*(d*x+c)/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-4/5*e^2*(d*x+c)^3/b^2/d /(a+b*arcsinh(d*x+c))^(3/2)+1/15*e^2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/ 2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d-1/15*e^2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^ (1/2))*Pi^(1/2)/b^(7/2)/d/exp(a/b)-3/5*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arc sinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/d+3/5*e^2*erfi(3^(1/2 )*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(3*a/b )-2/5*e^2*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(5/2)-16/ 15*e^2*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)-24/5*e^2*(d*x+ c)^2*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)
Time = 1.20 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.16 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {e^2 \left (3 b^2 e^{\text {arcsinh}(c+d x)}+e^{-\text {arcsinh}(c+d x)} \left (4 a^2-2 a b+3 b^2+2 (4 a-b) b \text {arcsinh}(c+d x)+4 b^2 \text {arcsinh}(c+d x)^2-4 e^{\frac {a}{b}+\text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x))^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-3 \left (b^2 e^{3 \text {arcsinh}(c+d x)}+2 e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (6 a+b+6 b \text {arcsinh}(c+d x))+6 \sqrt {3} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )+2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{\frac {a}{b}+\text {arcsinh}(c+d x)} (2 a+b+2 b \text {arcsinh}(c+d x))+2 b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )-3 e^{-3 \text {arcsinh}(c+d x)} \left (b^2+2 (a+b \text {arcsinh}(c+d x)) \left (6 a-b+6 b \text {arcsinh}(c+d x)-6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )\right )}{60 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]
(e^2*(3*b^2*E^ArcSinh[c + d*x] + (4*a^2 - 2*a*b + 3*b^2 + 2*(4*a - b)*b*Ar cSinh[c + d*x] + 4*b^2*ArcSinh[c + d*x]^2 - 4*E^(a/b + ArcSinh[c + d*x])*S qrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + Ar cSinh[c + d*x]])/E^ArcSinh[c + d*x] - 3*(b^2*E^(3*ArcSinh[c + d*x]) + (2*( a + b*ArcSinh[c + d*x])*(E^(3*(a/b + ArcSinh[c + d*x]))*(6*a + b + 6*b*Arc Sinh[c + d*x]) + 6*Sqrt[3]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1 /2, (-3*(a + b*ArcSinh[c + d*x]))/b]))/E^((3*a)/b)) + (2*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b *(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x ])/b)]))/E^(a/b) - (3*(b^2 + 2*(a + b*ArcSinh[c + d*x])*(6*a - b + 6*b*Arc Sinh[c + d*x] - 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSin h[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x] ))/b])))/E^(3*ArcSinh[c + d*x])))/(60*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2) )
Result contains complex when optimal does not.
Time = 2.12 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6274, 27, 6194, 6233, 6188, 6193, 2009, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634}
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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {6 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {e^2 \left (\frac {6 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^2 \left (\frac {6 \left (\frac {2 \left (\frac {2 \int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-i \int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {6 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
(e^2*((-2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(5*b*(a + b*ArcSinh[c + d*x]) ^(5/2)) + (4*((-2*(c + d*x))/(3*b*(a + b*ArcSinh[c + d*x])^(3/2)) + (2*((- 2*Sqrt[1 + (c + d*x)^2])/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + ((2*I)*((I/2)* Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - ((I/2 )*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b)))/b ^2))/(3*b)))/(5*b) + (6*((-2*(c + d*x)^3)/(3*b*(a + b*ArcSinh[c + d*x])^(3 /2)) + (2*((-2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + (2*((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/ Sqrt[b]])/8 - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcS inh[c + d*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/b^2))/b))/(5*b)))/d
3.3.24.3.1 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[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
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]
\[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**2/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b *asinh(c + d*x))*asinh(c + d*x) + 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asin h(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x) + I ntegral(d**2*x**2/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*a sinh(c + d*x))*asinh(c + d*x) + 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asinh( c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x) + Int egral(2*c*d*x/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh (c + d*x))*asinh(c + d*x) + 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]