Integrand size = 25, antiderivative size = 531 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}-\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}-\frac {9 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d} \]
-16/15*e^4*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-4/3*e^4*(d*x+c)^5/b^ 2/d/(a+b*arcsinh(d*x+c))^(3/2)-1/30*e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^ (1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d+1/30*e^4*erfi((a+b*arcsinh(d*x+c))^(1/2) /b^(1/2))*Pi^(1/2)/b^(7/2)/d/exp(a/b)+9/20*e^4*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^(7/2)/d-9/20*e^4*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)-5/12*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)/b^(7/2)/d+5/12*e^4*erfi(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/ 2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(5*a/b)-2/5*e^4*(d*x+c)^4*(1+(d* x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(5/2)-32/5*e^4*(d*x+c)^2*(1+(d*x+c) ^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)-40/3*e^4*(d*x+c)^4*(1+(d*x+c)^2 )^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)
Time = 2.25 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.32 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {e^4 \left (-6 b^2 e^{\text {arcsinh}(c+d x)}-3 b^2 e^{5 \text {arcsinh}(c+d x)}+e^{-\text {arcsinh}(c+d x)} \left (-8 a^2+4 a b-6 b^2-4 (4 a-b) b \text {arcsinh}(c+d x)-8 b^2 \text {arcsinh}(c+d x)^2+8 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 )-10 e^{-\frac {5 a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (10 a+b+10 b \text {arcsinh}(c+d x))+10 \sqrt {5} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )+9 \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 )-4 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 )+9 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 )-e^{-5 \text {arcsinh}(c+d x)} \left (3 b^2+10 (a+b \text {arcsinh}(c+d x)) \left (10 a-b+10 b \text {arcsinh}(c+d x)-10 \sqrt {5} e^{5 \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 {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )\right )}{240 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]
(e^4*(-6*b^2*E^ArcSinh[c + d*x] - 3*b^2*E^(5*ArcSinh[c + d*x]) + (-8*a^2 + 4*a*b - 6*b^2 - 4*(4*a - b)*b*ArcSinh[c + d*x] - 8*b^2*ArcSinh[c + d*x]^2 + 8*E^(a/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSin h[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]])/E^ArcSinh[c + d*x] - (10 *(a + b*ArcSinh[c + d*x])*(E^(5*(a/b + ArcSinh[c + d*x]))*(10*a + b + 10*b *ArcSinh[c + d*x]) + 10*Sqrt[5]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Ga mma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b]))/E^((5*a)/b) + 9*(b^2*E^(3*ArcS inh[c + d*x]) + (2*(a + b*ArcSinh[c + d*x])*(E^(3*(a/b + ArcSinh[c + d*x]) )*(6*a + b + 6*b*ArcSinh[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)) - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*Arc Sinh[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) + (9*(b^2 + 2*(a + b*ArcSinh[c + d*x] )*(6*a - b + 6*b*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])))/E^(3*ArcSinh[c + d*x]) - (3*b^2 + 10*(a + b*A rcSinh[c + d*x])*(10*a - b + 10*b*ArcSinh[c + d*x] - 10*Sqrt[5]*E^(5*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x]) *Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/E^(5*ArcSinh[c + d*x])))/(24 0*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))
Time = 1.54 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.31, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6274, 27, 6194, 6233, 6193, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {2 \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \left (\frac {2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {2 \left (\frac {10 \int \frac {(c+d x)^4}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \left (\frac {10 \left (\frac {2 \int \left (-\frac {5 \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {9 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^4 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{b}+\frac {8 \left (\frac {2 \left (\frac {2 \int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{b}-\frac {2 (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}+\frac {2 \left (\frac {10 \left (\frac {2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^4 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^5}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\) |
(e^4*((-2*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(5*b*(a + b*ArcSinh[c + d*x]) ^(5/2)) + (8*((-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*ArcSinh[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) + (2*((-2*(c + d*x)^ 5)/(3*b*(a + b*ArcSinh[c + d*x])^(3/2)) + (10*((-2*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + (2*(-1/16*(Sqrt[b]*E^(a/b)* Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]) + (3*Sqrt[b]*E^((3*a)/ b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/32 - (S qrt[b]*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/S qrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] )/(16*E^(a/b)) - (3*Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*E^((3*a)/b)) + (Sqrt[b]*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sq rt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*E^((5*a)/b))))/b^2))/(3*b)))/b)) /d
3.3.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] - 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_) + (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]
\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{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^{4} x^{4}}{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 {4 c d^{3} x^{3}}{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 {6 c^{2} 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 {4 c^{3} 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**4*(Integral(c**4/(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**4*x**4/(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(4*c*d**3*x**3/(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(6*c**2*d**2*x**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)) *asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x ) + Integral(4*c**3*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) )
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]