3.148 \(\int x^2 (a+b \cosh ^{-1}(c x))^{5/2} \, dx\)

Optimal. Leaf size=337 \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}+\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{5 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c} \]

[Out]

(5*b^2*x*Sqrt[a + b*ArcCosh[c*x]])/(6*c^2) + (5*b^2*x^3*Sqrt[a + b*ArcCosh[c*x]])/36 - (5*b*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3
/2))/(18*c) + (x^3*(a + b*ArcCosh[c*x])^(5/2))/3 - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/S
qrt[b]])/(64*c^3) - (5*b^(5/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^
3) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*c^3*E^(a/b)) - (5*b^(5/2)*Sqrt[Pi/3]*Erf
i[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^3*E^((3*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 2.08993, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5664, 5759, 5718, 5654, 5781, 3307, 2180, 2204, 2205, 3312} \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}+\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{5 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(5*b^2*x*Sqrt[a + b*ArcCosh[c*x]])/(6*c^2) + (5*b^2*x^3*Sqrt[a + b*ArcCosh[c*x]])/36 - (5*b*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3
/2))/(18*c) + (x^3*(a + b*ArcCosh[c*x])^(5/2))/3 - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/S
qrt[b]])/(64*c^3) - (5*b^(5/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^
3) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*c^3*E^(a/b)) - (5*b^(5/2)*Sqrt[Pi/3]*Erf
i[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^3*E^((3*a)/b))

Rule 5664

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

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5654

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

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 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 x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{1}{6} (5 b c) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac{1}{12} \left (5 b^2\right ) \int x^2 \sqrt{a+b \cosh ^{-1}(c x)} \, dx-\frac{(5 b) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c}\\ &=\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^2\right ) \int \sqrt{a+b \cosh ^{-1}(c x)} \, dx}{6 c^2}-\frac{1}{72} \left (5 b^3 c\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{72 c^3}-\frac{\left (5 b^3\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{12 c}\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 \sqrt{a+b x}}+\frac{\cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{72 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{288 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{576 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{576 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{192 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{192 c^3}\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{5 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{5 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{288 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{288 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{96 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{96 c^3}\\ &=\frac{5 b^2 x \sqrt{a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 b^{5/2} e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}-\frac{15 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 b^{5/2} e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}\\ \end{align*}

Mathematica [B]  time = 10.694, size = 924, normalized size = 2.74 \[ \frac{e^{-\frac{3 a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right ) a^2}{72 c^3 \sqrt{-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}}+\frac{\sqrt{b} \left (9 \left (-12 \sqrt{b} \sqrt{\frac{c x-1}{c x+1}} \sqrt{a+b \cosh ^{-1}(c x)} (c x+1)+(2 a+3 b) \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right )+(2 a-3 b) \sqrt{\pi } \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{a}{b}\right )+\sinh \left (\frac{a}{b}\right )\right )+8 \sqrt{b} c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right )+(2 a+b) \sqrt{3 \pi } \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{3 a}{b}\right )-\sinh \left (\frac{3 a}{b}\right )\right )+(2 a-b) \sqrt{3 \pi } \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{3 a}{b}\right )+\sinh \left (\frac{3 a}{b}\right )\right )+12 \sqrt{b} \sqrt{a+b \cosh ^{-1}(c x)} \left (2 \cosh ^{-1}(c x) \cosh \left (3 \cosh ^{-1}(c x)\right )-\sinh \left (3 \cosh ^{-1}(c x)\right )\right )\right ) a}{144 c^3}-\frac{27 \left (-4 b \sqrt{a+b \cosh ^{-1}(c x)} \left (2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a-5 b \cosh ^{-1}(c x)\right )+b c x \left (4 \cosh ^{-1}(c x)^2+15\right )\right )+\sqrt{b} \left (4 a^2+12 b a+15 b^2\right ) \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right )+\sqrt{b} \left (4 a^2-12 b a+15 b^2\right ) \sqrt{\pi } \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{a}{b}\right )+\sinh \left (\frac{a}{b}\right )\right )\right )+\sqrt{b} \left (12 a^2+12 b a+5 b^2\right ) \sqrt{3 \pi } \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{3 a}{b}\right )-\sinh \left (\frac{3 a}{b}\right )\right )+\sqrt{b} \left (12 a^2-12 b a+5 b^2\right ) \sqrt{3 \pi } \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{3 a}{b}\right )+\sinh \left (\frac{3 a}{b}\right )\right )-12 b \sqrt{a+b \cosh ^{-1}(c x)} \left (b \left (12 \cosh ^{-1}(c x)^2+5\right ) \cosh \left (3 \cosh ^{-1}(c x)\right )+2 \left (a-5 b \cosh ^{-1}(c x)\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )\right )}{1728 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(a^2*Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x]] +
Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*
x]]*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*
(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]) + (a*Sqrt[b]*(9*(-12*Sqrt[
b]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCo
sh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*S
qrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sq
rt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a +
b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcCosh[c*x]]*(2*ArcCosh[c*x
]*Cosh[3*ArcCosh[c*x]] - Sinh[3*ArcCosh[c*x]])))/(144*c^3) - (27*(-4*b*Sqrt[a + b*ArcCosh[c*x]]*(2*Sqrt[(-1 +
c*x)/(1 + c*x)]*(1 + c*x)*(a - 5*b*ArcCosh[c*x]) + b*c*x*(15 + 4*ArcCosh[c*x]^2)) + Sqrt[b]*(4*a^2 + 12*a*b +
15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + Sqrt[b]*(4*a^2 - 12*a*b + 15
*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(12*a^2 + 12*a*b + 5*b
^2)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + Sqrt[b]*(12*
a^2 - 12*a*b + 5*b^2)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b
]) - 12*b*Sqrt[a + b*ArcCosh[c*x]]*(b*(5 + 12*ArcCosh[c*x]^2)*Cosh[3*ArcCosh[c*x]] + 2*(a - 5*b*ArcCosh[c*x])*
Sinh[3*ArcCosh[c*x]]))/(1728*c^3)

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccosh(c*x))^(5/2),x)

[Out]

int(x^2*(a+b*arccosh(c*x))^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)^(5/2)*x^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(x^2*(a+b*arccosh(c*x))^(5/2),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(x**2*(a+b*acosh(c*x))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccosh(c*x))^(5/2),x, algorithm="giac")

[Out]

sage0*x