Integrand size = 25, antiderivative size = 437 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}+\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d} \]
1/12*e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2) /d+1/12*e^4*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d/ex p(a/b)-3/8*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^(5/2)/d-3/8*e^4*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)+5/24*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^(5/2)/d+5/24* e^4*erfi(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(5 /2)/d/exp(5*a/b)-2/3*e^4*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d* x+c))^(3/2)-16/3*e^4*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))^(1/2)-20/3*e^4*( d*x+c)^5/b^2/d/(a+b*arcsinh(d*x+c))^(1/2)
Time = 2.77 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.26 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\frac {e^4 \left (-2 e^{\text {arcsinh}(c+d x)} (2 a+b+2 b \text {arcsinh}(c+d x))+e^{-\text {arcsinh}(c+d x)} \left (4 a-2 b+4 b \text {arcsinh}(c+d x)-4 e^{\frac {a}{b}+\text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+e^{-\frac {5 a}{b}} \left (-e^{5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (10 a+b+10 b \text {arcsinh}(c+d x))-10 \sqrt {5} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )+e^{-\frac {3 a}{b}} \left (3 e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (6 a+b+6 b \text {arcsinh}(c+d x))+18 \sqrt {3} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )-4 b e^{-\frac {a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+3 e^{-3 \text {arcsinh}(c+d x)} \left (-6 a+b-6 b \text {arcsinh}(c+d x)+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )+e^{-5 \text {arcsinh}(c+d x)} \left (10 a-b+10 b \text {arcsinh}(c+d x)-10 \sqrt {5} e^{5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{48 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}} \]
(e^4*(-2*E^ArcSinh[c + d*x]*(2*a + b + 2*b*ArcSinh[c + d*x]) + (4*a - 2*b + 4*b*ArcSinh[c + d*x] - 4*E^(a/b + 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]])/E^Ar cSinh[c + d*x] + (-(E^(5*(a/b + ArcSinh[c + d*x]))*(10*a + b + 10*b*ArcSin h[c + d*x])) - 10*Sqrt[5]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/ 2, (-5*(a + b*ArcSinh[c + d*x]))/b])/E^((5*a)/b) + (3*E^(3*(a/b + ArcSinh[ c + d*x]))*(6*a + b + 6*b*ArcSinh[c + d*x]) + 18*Sqrt[3]*b*(-((a + b*ArcSi nh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b])/E^((3* a)/b) - (4*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*Arc Sinh[c + d*x])/b)])/E^(a/b) + (3*(-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*Arc Sinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/E^(3*ArcSinh[c + d*x]) + (10*a - b + 10*b*ArcSinh[c + d*x] - 10*Sqrt[5]*E^(5*(a/b + ArcS inh[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])))/(48*b^2*d* (a + b*ArcSinh[c + d*x])^(3/2))
Time = 1.59 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.39, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6274, 27, 6194, 6233, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}+\frac {10 \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {10 \left (\frac {10 \int \frac {(c+d x)^4}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {e^4 \left (\frac {10 \left (\frac {10 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {8 \left (\frac {6 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \left (\frac {6 \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {10 \left (\frac {10 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {8 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}+\frac {10 \left (\frac {10 \left (\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^5}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{d}\) |
(e^4*((-2*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(3*b*(a + b*ArcSinh[c + d*x]) ^(3/2)) + (8*((-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]]) + (S qrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/S qrt[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))/(3*b) + (10*((-2*(c + d*x)^5)/(b*Sq rt[a + b*ArcSinh[c + d*x]]) + (10*((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/16 - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sq rt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/32 + (Sqrt[b]*E^((5*a)/b)*Sq rt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/32 + (Sqrt[b ]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) - (Sqr t[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32* E^((3*a)/b)) + (Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d* x]])/Sqrt[b]])/(32*E^((5*a)/b))))/b^2))/(3*b)))/d
3.3.16.3.1 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[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*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]
\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(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**4*x**4/(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(4*c*d**3*x**3/(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(6*c**2*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(4 *c**3*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*asinh(c + d*x))*asinh(c + d*x)**2), x ))
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]