3.629 \(\int (d+e x^2)^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx\)

Optimal. Leaf size=672 \[ -\frac{\sqrt{\pi } \sqrt{b} d e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\sqrt{\pi } \sqrt{b} d e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}+\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}-\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}+\frac{\sqrt{\pi } \sqrt{b} d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)} \]

[Out]

d^2*x*Sqrt[a + b*ArcSinh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcSinh[c*x]])/5
+ (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*
Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3) + (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(32*c^5) + (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^
3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5) + (Sqrt[b]*
e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*
Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sq
rt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) + (Sqrt[b]*e^2*Sqr
t[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[
(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 1.8788, antiderivative size = 672, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5706, 5653, 5779, 3308, 2180, 2204, 2205, 5663, 3312} \[ -\frac{\sqrt{\pi } \sqrt{b} d e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\sqrt{\pi } \sqrt{b} d e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}+\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}-\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}+\frac{\sqrt{\pi } \sqrt{b} d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*x*Sqrt[a + b*ArcSinh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcSinh[c*x]])/5
+ (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*
Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3) + (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(32*c^5) + (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^
3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5) + (Sqrt[b]*
e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*
Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sq
rt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sq
rt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) + (Sqrt[b]*e^2*Sqr
t[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[
(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*a)/b))

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

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 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 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 \left (d+e x^2\right )^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx &=\int \left (d^2 \sqrt{a+b \sinh ^{-1}(c x)}+2 d e x^2 \sqrt{a+b \sinh ^{-1}(c x)}+e^2 x^4 \sqrt{a+b \sinh ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \sqrt{a+b \sinh ^{-1}(c x)} \, dx+(2 d e) \int x^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx+e^2 \int x^4 \sqrt{a+b \sinh ^{-1}(c x)} \, dx\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{1}{2} \left (b c d^2\right ) \int \frac{x}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx-\frac{1}{3} (b c d e) \int \frac{x^3}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx-\frac{1}{10} \left (b c e^2\right ) \int \frac{x^5}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}-\frac{(b d e) \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^3}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh ^5(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac{(i b d e) \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 x)\right )}{3 c^3}+\frac{\left (i b e^2\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 x)\right )}{10 c^5}\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{d^2 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac{d^2 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac{(b d e) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}+\frac{(b d e) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{160 c^5}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^5}\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} d^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{(b d e) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}-\frac{(b d e) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}-\frac{(b d e) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}+\frac{(b d e) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{320 c^5}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{320 c^5}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^5}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^5}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^5}\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} d^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{(d e) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{12 c^3}-\frac{(d e) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{12 c^3}-\frac{(d e) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 c^3}+\frac{(d e) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 c^3}+\frac{e^2 \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{160 c^5}-\frac{e^2 \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{160 c^5}-\frac{e^2 \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{32 c^5}+\frac{e^2 \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{32 c^5}+\frac{e^2 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{16 c^5}-\frac{e^2 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{16 c^5}\\ &=d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} d e e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{b} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}+\frac{\sqrt{b} d e e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{\sqrt{b} e^2 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}+\frac{\sqrt{b} e^2 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}-\frac{\sqrt{b} d^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{\sqrt{b} d e e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}-\frac{\sqrt{b} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^5}-\frac{\sqrt{b} d e e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\sqrt{b} e^2 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^5}-\frac{\sqrt{b} e^2 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{320 c^5}\\ \end{align*}

Mathematica [A]  time = 6.17694, size = 535, normalized size = 0.8 \[ -\frac{b e^{-\frac{5 a}{b}} \left (450 e^{\frac{6 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right ) \left (b e \left (4 c^2 d-e\right ) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}+8 a c^4 d^2 \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}\right )+450 e^{\frac{4 a}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right ) \left (b e \left (e-4 c^2 d\right ) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}+8 a c^4 d^2 \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}\right )+25 \sqrt{3} b e e^{\frac{2 a}{b}} \left (8 c^2 d-3 e\right ) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b e e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \left (25 \sqrt{3} \left (8 c^2 d-3 e\right ) \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+9 \sqrt{5} e e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )+9 \sqrt{5} b e^2 \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}} \text{Gamma}\left (\frac{3}{2},-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{7200 c^5 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcSinh[c*x]] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[a/b + ArcSinh[c*x]]
+ b*(4*c^2*d - e)*e*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, a/b + Arc
Sinh[c*x]] + 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*Gamma[3/2, (-5*(a +
b*ArcSinh[c*x]))/b] + 25*Sqrt[3]*b*(8*c^2*d - 3*e)*e*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSin
h[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] + 450*E^((4*a)/b)*(8*a*c^4*d^2*Sqrt[-((a + b*ArcSinh[c
*x])/b)] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*e*(-4*c^2*d + e)*Sqrt[a/b + ArcSinh[c*
x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcSinh[c*x])/b)] - b*e*E^((8*a)/b)*Sqrt[-((a + b
*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d - 3*e)*Gamma[3/2, (3*(a + b*ArcSin
h[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcSinh[c*x]))/b])))/(7200*c^5*E^((5*a)/b)*(a + b*A
rcSinh[c*x])^(3/2))

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Maple [F]  time = 0.35, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) ^{2}\sqrt{a+b{\it Arcsinh} \left ( cx \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x)

[Out]

int((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^2*sqrt(b*arcsinh(c*x) + a), 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((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*asinh(c*x))**(1/2),x)

[Out]

Integral(sqrt(a + b*asinh(c*x))*(d + e*x**2)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*sqrt(b*arcsinh(c*x) + a), x)