Integrand size = 25, antiderivative size = 321 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}-\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d} \]
-1/6*e^2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2) /d-1/6*e^2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d/exp (a/b)+1/2*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3 ^(1/2)*Pi^(1/2)/b^(5/2)/d+1/2*e^2*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/ b^(1/2))*3^(1/2)*Pi^(1/2)/b^(5/2)/d/exp(3*a/b)-2/3*e^2*(d*x+c)^2*(1+(d*x+c )^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(3/2)-8/3*e^2*(d*x+c)/b^2/d/(a+b*arcsi nh(d*x+c))^(1/2)-4*e^2*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))^(1/2)
Time = 1.15 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.21 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\frac {e^2 e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \left (2 e^{\frac {4 a}{b}+3 \text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )-6 \sqrt {3} b e^{3 \text {arcsinh}(c+d x)} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+2 b e^{\frac {2 a}{b}+3 \text {arcsinh}(c+d x)} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-e^{\frac {3 a}{b}} \left (\left (-1+e^{2 \text {arcsinh}(c+d x)}\right ) \left (b \left (-1+e^{4 \text {arcsinh}(c+d x)}\right )+a \left (6+4 e^{2 \text {arcsinh}(c+d x)}+6 e^{4 \text {arcsinh}(c+d x)}\right )+2 b \left (3+2 e^{2 \text {arcsinh}(c+d x)}+3 e^{4 \text {arcsinh}(c+d x)}\right ) \text {arcsinh}(c+d x)\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{12 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}} \]
(e^2*(2*E^((4*a)/b + 3*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 6*Sqrt[3]*b*E^(3 *ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 2*b*E^((2*a)/b + 3*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] - E^((3*a)/b)*((-1 + E^(2*ArcSinh[c + d*x]))*(b*(-1 + E^(4*ArcSinh[c + d*x]) ) + a*(6 + 4*E^(2*ArcSinh[c + d*x]) + 6*E^(4*ArcSinh[c + d*x])) + 2*b*(3 + 2*E^(2*ArcSinh[c + d*x]) + 3*E^(4*ArcSinh[c + d*x]))*ArcSinh[c + d*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b])))/(12*b^2 *d*E^(3*(a/b + ArcSinh[c + d*x]))*(a + b*ArcSinh[c + d*x])^(3/2))
Time = 1.85 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.24, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6274, 27, 6194, 6233, 6189, 3042, 3788, 26, 2611, 2633, 2634, 6195, 5971, 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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}+\frac {2 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6189 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {6 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {6 \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
(e^2*((-2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(3*b*(a + b*ArcSinh[c + d*x]) ^(3/2)) + (4*((-2*(c + d*x))/(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]])/2 + (Sqrt[b ]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*E^(a/b))))/b^2)) /(3*b) + (2*((-2*(c + d*x)^3)/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + (6*(-1/8* (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]) + (Sq rt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sq rt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/ (8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x ]])/Sqrt[b]])/(8*E^((3*a)/b))))/b^2))/b))/d
3.3.18.3.1 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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
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_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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_)*(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_)*((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_) + (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]
\[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**2/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*b*sqrt(a + b*as inh(c + d*x))*asinh(c + d*x) + b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d *x)**2), x) + Integral(d**2*x**2/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*b* sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*sqrt(a + b*asinh(c + d*x) )*asinh(c + d*x)**2), x) + Integral(2*c*d*x/(a**2*sqrt(a + b*asinh(c + d*x )) + 2*a*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*sqrt(a + b*asi nh(c + d*x))*asinh(c + d*x)**2), x))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]