3.2.98 \(\int (c e+d e x)^4 (a+b \sinh ^{-1}(c+d x))^{7/2} \, dx\) [198]

Optimal. Leaf size=835 \[ -\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {105 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {105 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d} \]

[Out]

14/15*b^2*e^4*(d*x+c)*(a+b*arcsinh(d*x+c))^(3/2)/d-7/45*b^2*e^4*(d*x+c)^3*(a+b*arcsinh(d*x+c))^(3/2)/d+7/100*b
^2*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^(3/2)/d+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^(7/2)/d+21/320000*b^(7/2)
*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)/d+21/320000*b^(7/2)*e^4*erfi(
5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/d/exp(5*a/b)-35/13824*b^(7/2)*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)/d-35/13824*b^(7/2)*e^4*erfi(3^(1/2)*(a+b*arcsinh(
d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/d/exp(3*a/b)+105/256*b^(7/2)*e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/
2)/b^(1/2))*Pi^(1/2)/d+105/256*b^(7/2)*e^4*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)-28/75*
b*e^4*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d+14/75*b*e^4*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*
x+c)^2)^(1/2)/d-7/50*b*e^4*(d*x+c)^4*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d-1813/1125*b^3*e^4*(1+(d*
x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d+119/1125*b^3*e^4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))
^(1/2)/d-21/1000*b^3*e^4*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 2.09, antiderivative size = 835, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5859, 12, 5777, 5812, 5798, 5772, 5774, 3388, 2211, 2236, 2235, 5780, 5556} \begin {gather*} \frac {e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} (c+d x)^5}{5 d}+\frac {7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^5}{100 d}-\frac {7 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^4}{50 d}-\frac {21 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)} (c+d x)^4}{1000 d}-\frac {7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^3}{45 d}+\frac {14 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^2}{75 d}+\frac {119 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)} (c+d x)^2}{1125 d}+\frac {14 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)}{15 d}-\frac {28 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {105 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {119 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {105 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {119 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {1813 b^3 e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(-1813*b^3*e^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(1125*d) + (119*b^3*e^4*(c + d*x)^2*Sqrt[1
+ (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(1125*d) - (21*b^3*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a +
b*ArcSinh[c + d*x]])/(1000*d) + (14*b^2*e^4*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/(15*d) - (7*b^2*e^4*(c
+ d*x)^3*(a + b*ArcSinh[c + d*x])^(3/2))/(45*d) + (7*b^2*e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(3/2))/(100*
d) - (28*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(75*d) + (14*b*e^4*(c + d*x)^2*Sqrt[1 + (
c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(75*d) - (7*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh
[c + d*x])^(5/2))/(50*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(7/2))/(5*d) + (105*b^(7/2)*e^4*E^(a/b)*S
qrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(256*d) - (119*b^(7/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt
[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(18000*d) - (21*b^(7/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sq
rt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d) + (21*b^(7/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a +
b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d) + (105*b^(7/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b
]])/(256*d*E^(a/b)) - (119*b^(7/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(18000
*d*E^((3*a)/b)) - (21*b^(7/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d*E^
((3*a)/b)) + (21*b^(7/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d*E^((5*a
)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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]

Rule 5556

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]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5812

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] + (-Dist[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] - Dist[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
]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps

\begin {align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac {\left (7 b e^4\right ) \text {Subst}\left (\int \frac {x^5 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{10 d}\\ &=-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (14 b e^4\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (7 b^2 e^4\right ) \text {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{20 d}\\ &=\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac {\left (28 b e^4\right ) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{75 d}-\frac {\left (7 b^2 e^4\right ) \text {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{15 d}-\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5 \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{200 d}\\ &=-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (14 b^2 e^4\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{15 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{250 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{30 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{2000 d}\\ &=\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{125 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{45 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{500 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{180 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{16 \sqrt {a+b x}}+\frac {\cosh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{500 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{250 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{180 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{90 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{10 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{250 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{90 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{10 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16000 d}-\frac {\left (63 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{500 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{180 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{500 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{500 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{180 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{180 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{20 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{20 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}+\frac {\left (21 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}-\frac {\left (63 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}-\frac {\left (63 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{720 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{720 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{16000 d}+\frac {\left (21 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{16000 d}-\frac {\left (63 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}-\frac {\left (63 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{250 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{250 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{90 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{90 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{10 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{10 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}+\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}-\frac {\left (7 b^4 e^4\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {116221 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{288000 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {116221 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{288000 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{720 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{720 d}+\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{720 d}-\frac {\left (7 b^3 e^4\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{720 d}\\ &=-\frac {1813 b^3 e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac {119 b^3 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac {21 b^3 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac {14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac {7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac {28 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac {14 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac {7 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac {105 b^{7/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {105 b^{7/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {119 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{18000 d}-\frac {21 b^{7/2} e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}+\frac {21 b^{7/2} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64000 d}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 343, normalized size = 0.41 \begin {gather*} \frac {b e^4 e^{-\frac {5 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (506250 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {9}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+81 \sqrt {5} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {9}{2},-\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-3125 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {9}{2},-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+506250 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {9}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )-3125 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {9}{2},\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+81 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {9}{2},\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{8100000 d \left (-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(b*e^4*(a + b*ArcSinh[c + d*x])^(5/2)*(506250*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, a/b +
ArcSinh[c + d*x]] + 81*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 312
5*Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 506250*E^((4*
a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, -((a + b*ArcSinh[c + d*x])/b)] - 3125*Sqrt[3]*E^((8*a)/b)*Sqrt[-
((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (3*(a + b*ArcSinh[c + d*x]))/b] + 81*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a +
b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(8100000*d*E^((5*a)/b)*(-((a + b*ArcSinh
[c + d*x])^2/b^2))^(3/2))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{4} \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {7}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(7/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^4*(b*arcsinh(d*x + c) + a)^(7/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4*(a+b*asinh(d*x+c))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7315 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^(7/2), x)

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