\(\int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx\) [148]

Optimal result

Integrand size = 16, antiderivative size = 337 \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {5 b^2 x \sqrt {a+b \text {arccosh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3} \]

[Out]

1/3*x^3*(a+b*arccosh(c*x))^(5/2)-5/1728*b^(5/2)*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/
2)*Pi^(1/2)/c^3-5/1728*b^(5/2)*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-
15/64*b^(5/2)*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3-15/64*b^(5/2)*erfi((a+b*arccosh(c*x)
)^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)-5/9*b*(a+b*arccosh(c*x))^(3/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-5/18*b*x
^2*(a+b*arccosh(c*x))^(3/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+5/6*b^2*x*(a+b*arccosh(c*x))^(1/2)/c^2+5/36*b^2*x^3*
(a+b*arccosh(c*x))^(1/2)

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5884, 5939, 5915, 5879, 5953, 3388, 2211, 2236, 2235, 3393} \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {5 b^2 x \sqrt {a+b \text {arccosh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{9 c^3}+\frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{5/2}-\frac {5 b x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{18 c} \]

[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 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 3393

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])
)

Rule 5879

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 5884

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5939

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

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(924\) vs. \(2(337)=674\).

Time = 8.75 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.74 \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {a^2 e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}}+\frac {a \sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 \sqrt {b} c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(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 \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(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 \text {arccosh}(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 \text {arccosh}(c x)} (2 \text {arccosh}(c x) \cosh (3 \text {arccosh}(c x))-\sinh (3 \text {arccosh}(c x)))\right )}{144 c^3}-\frac {27 \left (-4 b \sqrt {a+b \text {arccosh}(c x)} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a-5 b \text {arccosh}(c x))+b c x \left (15+4 \text {arccosh}(c x)^2\right )\right )+\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(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 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(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 a b+5 b^2\right ) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(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 a b+5 b^2\right ) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(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 \text {arccosh}(c x)} \left (b \left (5+12 \text {arccosh}(c x)^2\right ) \cosh (3 \text {arccosh}(c x))+2 (a-5 b \text {arccosh}(c x)) \sinh (3 \text {arccosh}(c x))\right )}{1728 c^3} \]

[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)

Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b \text {arccosh}(c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \]

[In]

int(x^2*(a + b*acosh(c*x))^(5/2),x)

[Out]

int(x^2*(a + b*acosh(c*x))^(5/2), x)