3.192 \(\int (c e+d e x)^4 (a+b \sinh ^{-1}(c+d x))^{5/2} \, dx\)

Optimal. Leaf size=701 \[ \frac{15 \sqrt{\pi } b^{5/2} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1280 d}-\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{240 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{6400 d}-\frac{15 \sqrt{\pi } b^{5/2} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1280 d}+\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{240 d}-\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{6400 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac{b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d} \]

[Out]

(2*b^2*e^4*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(5*d) - (b^2*e^4*(c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/
(15*d) + (3*b^2*e^4*(c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/(100*d) - (4*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*
ArcSinh[c + d*x])^(3/2))/(15*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(
15*d) - (b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(10*d) + (e^4*(c + d*x)^5*(a
+ b*ArcSinh[c + d*x])^(5/2))/(5*d) + (15*b^(5/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]
])/(128*d) - (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d)
- (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d) + (3*b^(5/
2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(6400*d) - (15*b^(5/2)*e^4*
Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (b^(5/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*S
qrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d*E^((3*a)/b)) + (b^(5/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*
ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d*E^((3*a)/b)) - (3*b^(5/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh
[c + d*x]])/Sqrt[b]])/(6400*d*E^((5*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 2.20313, antiderivative size = 701, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {5865, 12, 5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ \frac{15 \sqrt{\pi } b^{5/2} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1280 d}-\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{240 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{6400 d}-\frac{15 \sqrt{\pi } b^{5/2} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1280 d}+\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{240 d}-\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{6400 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac{b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

(2*b^2*e^4*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(5*d) - (b^2*e^4*(c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/
(15*d) + (3*b^2*e^4*(c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/(100*d) - (4*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*
ArcSinh[c + d*x])^(3/2))/(15*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(
15*d) - (b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(10*d) + (e^4*(c + d*x)^5*(a
+ b*ArcSinh[c + d*x])^(5/2))/(5*d) + (15*b^(5/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]
])/(128*d) - (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d)
- (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d) + (3*b^(5/
2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(6400*d) - (15*b^(5/2)*e^4*
Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (b^(5/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*S
qrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d*E^((3*a)/b)) + (b^(5/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*
ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d*E^((3*a)/b)) - (3*b^(5/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh
[c + d*x]])/Sqrt[b]])/(6400*d*E^((5*a)/b))

Rule 5865

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
]

Rule 12

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

Rule 5663

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 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*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}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 5653

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 5779

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

Rule 3308

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

Rule 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (2 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{5 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int x^4 \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{20 d}\\ &=\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac{\left (4 b e^4\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{15 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int x^2 \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{5 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{200 d}\\ &=-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (2 b^2 e^4\right ) \operatorname{Subst}\left (\int \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{5 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh ^5(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{30 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (3 i b^3 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{5 i \sinh (x)}{8 \sqrt{a+b x}}-\frac{5 i \sinh (3 x)}{16 \sqrt{a+b x}}+\frac{i \sinh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{30 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{5 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (i b^3 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{a+b x}}-\frac{i \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{30 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{320 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6400 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6400 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1280 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1280 d}+\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}-\frac{\left (3 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{120 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{40 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{10 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{10 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3200 d}-\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3200 d}-\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{640 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{640 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{320 d}-\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{320 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{240 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{240 d}+\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{80 d}-\frac{\left (b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{80 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{67 b^{5/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{640 d}-\frac{b^{5/2} 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 )}{1280 d}+\frac{3 b^{5/2} 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 )}{6400 d}-\frac{67 b^{5/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{640 d}+\frac{b^{5/2} 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 )}{1280 d}-\frac{3 b^{5/2} 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 )}{6400 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{120 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{120 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{40 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{40 d}\\ &=\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac{15 b^{5/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}-\frac{b^{5/2} 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 )}{240 d}-\frac{b^{5/2} 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 )}{1280 d}+\frac{3 b^{5/2} 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 )}{6400 d}-\frac{15 b^{5/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{b^{5/2} 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 )}{240 d}+\frac{b^{5/2} 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 )}{1280 d}-\frac{3 b^{5/2} 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 )}{6400 d}\\ \end{align*}

Mathematica [A]  time = 0.698151, size = 342, normalized size = 0.49 \[ -\frac{e^4 e^{-\frac{5 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (-33750 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{7}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+27 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{7}{2},-\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-625 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{7}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+33750 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{7}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+625 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{7}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-27 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{7}{2},\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{540000 d \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

-(e^4*(a + b*ArcSinh[c + d*x])^(5/2)*(-33750*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, a/b +
ArcSinh[c + d*x]] + 27*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 625*
Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 33750*E^((4*a)/
b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, -((a + b*ArcSinh[c + d*x])/b)] + 625*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a
+ b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (3*(a + b*ArcSinh[c + d*x]))/b] - 27*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*A
rcSinh[c + d*x])/b)]*Gamma[7/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(540000*d*E^((5*a)/b)*(-((a + b*ArcSinh[c +
d*x])^2/b^2))^(3/2))

________________________________________________________________________________________

Maple [F]  time = 0.323, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{4} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(5/2),x)

[Out]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(5/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{4}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^(5/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4*(a+b*asinh(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{4}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^(5/2), x)