3.186 \(\int (c e+d e x)^4 (a+b \sinh ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=601 \[ \frac{3 \sqrt{\pi } b^{3/2} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}+\frac{3 \sqrt{\pi } b^{3/2} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac{3 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d} \]

[Out]

(-4*b*e^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)
^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) - (3*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d
*x]])/(50*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(3/2))/(5*d) + (3*b^(3/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sq
rt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d) - (b^(3/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcS
inh[c + d*x]])/Sqrt[b]])/(200*d) - (3*b^(3/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d
*x]])/Sqrt[b]])/(3200*d) + (3*b^(3/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sq
rt[b]])/(3200*d) + (3*b^(3/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d*E^(a/b)) - (b^(3/
2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(200*d*E^((3*a)/b)) - (3*b^(3/2)*e^4*S
qrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d*E^((3*a)/b)) + (3*b^(3/2)*e^4*Sqrt[Pi/
5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d*E^((5*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 1.66167, antiderivative size = 601, normalized size of antiderivative = 1., number of steps used = 43, 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, 5657, 3307, 2180, 2205, 2204, 5669, 5448} \[ \frac{3 \sqrt{\pi } b^{3/2} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}+\frac{3 \sqrt{\pi } b^{3/2} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3200 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac{3 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(3/2),x]

[Out]

(-4*b*e^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)
^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) - (3*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d
*x]])/(50*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(3/2))/(5*d) + (3*b^(3/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sq
rt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d) - (b^(3/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcS
inh[c + d*x]])/Sqrt[b]])/(200*d) - (3*b^(3/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d
*x]])/Sqrt[b]])/(3200*d) + (3*b^(3/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sq
rt[b]])/(3200*d) + (3*b^(3/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d*E^(a/b)) - (b^(3/
2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(200*d*E^((3*a)/b)) - (3*b^(3/2)*e^4*S
qrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d*E^((3*a)/b)) + (3*b^(3/2)*e^4*Sqrt[Pi/
5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*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 5657

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 3307

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 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 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 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 5669

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

Rule 5448

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]

Rubi steps

\begin{align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac{\left (3 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{10 d}\\ &=-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{\left (6 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{100 d}\\ &=\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac{\left (4 b e^4\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{100 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{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 )}{100 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{25 d}+\frac{\left (2 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{\left (2 b e^4\right ) \operatorname{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 )}{25 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{800 d}-\frac{\left (9 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}-\frac{\left (b^2 e^4\right ) \operatorname{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 )}{25 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{\left (b e^4\right ) \operatorname{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 )}{25 d}+\frac{\left (b e^4\right ) \operatorname{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 )}{25 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}+\frac{\left (3 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}-\frac{\left (9 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}-\frac{\left (9 b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{100 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{100 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{\left (3 b 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 )}{1600 d}+\frac{\left (3 b 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 )}{1600 d}+\frac{\left (3 b 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 )}{800 d}+\frac{\left (3 b 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 )}{800 d}-\frac{\left (9 b 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 )}{1600 d}-\frac{\left (9 b 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 )}{1600 d}+\frac{\left (2 b 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 )}{25 d}+\frac{\left (2 b 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 )}{25 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}-\frac{\left (b^2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{67 b^{3/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1600 d}-\frac{3 b^{3/2} 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 )}{3200 d}+\frac{3 b^{3/2} 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 )}{3200 d}+\frac{67 b^{3/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1600 d}-\frac{3 b^{3/2} 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 )}{3200 d}+\frac{3 b^{3/2} 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 )}{3200 d}-\frac{\left (b 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 )}{100 d}+\frac{\left (b 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 )}{100 d}+\frac{\left (b 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 )}{100 d}-\frac{\left (b 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 )}{100 d}\\ &=-\frac{4 b e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac{2 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{3 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac{3 b^{3/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{b^{3/2} 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 )}{200 d}-\frac{3 b^{3/2} 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 )}{3200 d}+\frac{3 b^{3/2} 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 )}{3200 d}+\frac{3 b^{3/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{b^{3/2} 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 )}{200 d}-\frac{3 b^{3/2} 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 )}{3200 d}+\frac{3 b^{3/2} 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 )}{3200 d}\\ \end{align*}

Mathematica [A]  time = 0.467546, size = 343, normalized size = 0.57 \[ -\frac{b e^4 e^{-\frac{5 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (2250 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{5}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+9 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{5}{2},-\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-125 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{5}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+2250 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{5}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-125 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{5}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+9 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{5}{2},\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{36000 d \sqrt{-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(3/2),x]

[Out]

-(b*e^4*Sqrt[a + b*ArcSinh[c + d*x]]*(2250*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, a/b + Ar
cSinh[c + d*x]] + 9*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 125*Sqr
t[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 2250*E^((4*a)/b)*S
qrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, -((a + b*ArcSinh[c + d*x])/b)] - 125*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*
ArcSinh[c + d*x])/b)]*Gamma[5/2, (3*(a + b*ArcSinh[c + d*x]))/b] + 9*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*ArcSin
h[c + d*x])/b)]*Gamma[5/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(36000*d*E^((5*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*
x])^2/b^2)])

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Maple [F]  time = 0.332, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{4} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{3}{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))^(3/2),x)

[Out]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^(3/2),x)

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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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^(3/2), x)

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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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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))**(3/2),x)

[Out]

Timed out

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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{3}{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))^(3/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^(3/2), x)