\(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) [187]

Optimal result

Integrand size = 29, antiderivative size = 841 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)} \]

[Out]

5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/8*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2+35/9216
*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+215/13824*b^2*d^2*x^3*(-c^2*d*x^2+d)^(1/2)-5/864*b^2*c^2*d^2*x^5*(-c^2*d*x
^2+d)^(1/2)+73/12288*b^2*d^2*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2/(-c*x+1)/(c*x+1)+73/18432*b^2*d^2*x^3*(-c
^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)-43/4608*b^2*c^2*d^2*x^5*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*
x+1)/(c*x+1)+1/256*b^2*c^4*d^2*x^7*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)-5/128*d^2*x*(a+b*arccosh
(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+35/9216*b^2*d^2*arcco
sh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/128*b*d^2*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)
^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/384*b*c*d^2*x^4*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/
(c*x+1)^(1/2)+17/144*b*c^3*d^2*x^6*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/32*b*
c^5*d^2*x^8*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/384*d^2*(a+b*arccosh(c*x))^3
*(-c^2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-73/12288*b^2*d^2*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x
^2-1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c*x+1)/(c*x+1)

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {5930, 5926, 5939, 5893, 5883, 92, 54, 102, 12, 5912, 14, 5921, 471, 272, 45, 534, 1281, 470, 327, 223, 212} \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {b c^5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^8}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^4 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^7}{256 (1-c x) (c x+1)}+\frac {17 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^6}{144 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{864} b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^5-\frac {43 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^5}{4608 (1-c x) (c x+1)}-\frac {59 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^4}{384 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^3+\frac {5}{48} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^3+\frac {5}{64} d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^3+\frac {215 b^2 d^2 \sqrt {d-c^2 d x^2} x^3}{13824}+\frac {73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^3}{18432 (1-c x) (c x+1)}+\frac {5 b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^2}{128 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x}{128 c^2}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} x}{9216 c^2}+\frac {73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{12288 c^2 (1-c x) (c x+1)}-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {73 b^2 d^2 \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{12288 c^3 (1-c x) (c x+1)} \]

[In]

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

[Out]

(35*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/(9216*c^2) + (215*b^2*d^2*x^3*Sqrt[d - c^2*d*x^2])/13824 - (5*b^2*c^2*d^2*x
^5*Sqrt[d - c^2*d*x^2])/864 + (73*b^2*d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(12288*c^2*(1 - c*x)*(1 + c*x))
+ (73*b^2*d^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(18432*(1 - c*x)*(1 + c*x)) - (43*b^2*c^2*d^2*x^5*(1 - c
^2*x^2)*Sqrt[d - c^2*d*x^2])/(4608*(1 - c*x)*(1 + c*x)) + (b^2*c^4*d^2*x^7*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/
(256*(1 - c*x)*(1 + c*x)) + (35*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(9216*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^
2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(384*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*x^6*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(144*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x]))/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(1
28*c^2) + (5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/64 + (5*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*Ar
cCosh[c*x])^2)/48 + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/8 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^3)/(384*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (73*b^2*d^2*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*
ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(12288*c^3*(1 - c*x)*(1 + c*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 327

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

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 5883

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

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5912

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

Rule 5921

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5926

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/(f*(m + 2))), x] + (-Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2
]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]
- Dist[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(f*x)^(m + 1)*(a + b*Arc
Cosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m,
-2] || EqQ[n, 1])

Rule 5930

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} (5 d) \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{16} \left (5 d^2\right ) \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 (a+b \text {arccosh}(c x)) \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-3+2 c^2 x^2\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{512} b^2 d^2 x^3 \sqrt {d-c^2 d x^2}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{512 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-3+2 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{96 (-1+c x) (1+c x)} \\ & = -\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{256 c^2}+\frac {5}{512} b^2 d^2 x^3 \sqrt {d-c^2 d x^2}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{512 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{256 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{216 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (48 c^2-43 c^4 x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{768 (-1+c x) (1+c x)} \\ & = -\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{864 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1024 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{4608 (-1+c x) (1+c x)} \\ & = -\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1024 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{6144 (-1+c x) (1+c x)} \\ & = \frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1024 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{576 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{12288 c^2 (-1+c x) (1+c x)} \\ & = \frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^2 (-1+c x) (1+c x)} \\ & = \frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 5.92 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.08 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d^2 \left (34560 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+34560 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^7 x^7 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^8 x^8 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+11520 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3+34560 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+34560 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+13824 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+3456 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-1536 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+216 a b \sqrt {d-c^2 d x^2} \cosh (8 \text {arccosh}(c x))-6912 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-864 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+256 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-27 b^2 \sqrt {d-c^2 d x^2} \sinh (8 \text {arccosh}(c x))+24 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (576 b \cosh (2 \text {arccosh}(c x))+144 b \cosh (4 \text {arccosh}(c x))-64 b \cosh (6 \text {arccosh}(c x))+9 b \cosh (8 \text {arccosh}(c x))-1152 a \sinh (2 \text {arccosh}(c x))-576 a \sinh (4 \text {arccosh}(c x))+384 a \sinh (6 \text {arccosh}(c x))-72 a \sinh (8 \text {arccosh}(c x)))-288 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-120 a+48 b \sinh (2 \text {arccosh}(c x))+24 b \sinh (4 \text {arccosh}(c x))-16 b \sinh (6 \text {arccosh}(c x))+3 b \sinh (8 \text {arccosh}(c x)))\right )}{884736 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

-1/884736*(d^2*(34560*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 34560*a^2*c^2*x^2*Sqrt[(-1 + c*
x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872
*a^2*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^5*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sq
rt[d - c^2*d*x^2] + 313344*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^7*x^7*Sqr
t[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^8*x^8*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2
] + 11520*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^3 + 34560*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sq
rt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 34560*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sq
rt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 13824*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 3456*a*b*S
qrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 1536*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 216*a*b*Sqrt[d -
c^2*d*x^2]*Cosh[8*ArcCosh[c*x]] - 6912*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 864*b^2*Sqrt[d - c^2*d*
x^2]*Sinh[4*ArcCosh[c*x]] + 256*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 27*b^2*Sqrt[d - c^2*d*x^2]*Sinh
[8*ArcCosh[c*x]] + 24*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*(576*b*Cosh[2*ArcCosh[c*x]] + 144*b*Cosh[4*ArcCosh[c*
x]] - 64*b*Cosh[6*ArcCosh[c*x]] + 9*b*Cosh[8*ArcCosh[c*x]] - 1152*a*Sinh[2*ArcCosh[c*x]] - 576*a*Sinh[4*ArcCos
h[c*x]] + 384*a*Sinh[6*ArcCosh[c*x]] - 72*a*Sinh[8*ArcCosh[c*x]]) - 288*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2*(
-120*a + 48*b*Sinh[2*ArcCosh[c*x]] + 24*b*Sinh[4*ArcCosh[c*x]] - 16*b*Sinh[6*ArcCosh[c*x]] + 3*b*Sinh[8*ArcCos
h[c*x]])))/(c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs. \(2(741)=1482\).

Time = 0.88 (sec) , antiderivative size = 2528, normalized size of antiderivative = 3.01

method	result	size
default	\(\text {Expression too large to display}\)	\(2528\)
parts	\(\text {Expression too large to display}\)	\(2528\)

[In]

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/
2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/128*a^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2
+d)^(1/2))+b^2*(-5/384*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^3*d^2+1/65536*(-d*(
c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+272*c^5*x^5-256*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*c^6*x^6-88*c^3*x^3+160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(c*x-1)^(1/2)*(c*x+1)^(1/
2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(32*arccosh(c*x)^2-8*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)-1/6912*(-
d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+38*c^3*x^3-48*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*c^4*x^4-6*c*x+18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(18*arccosh(c
*x)^2-6*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccos
h(c*x)^2-4*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/
2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(2*arccosh(c*x)^2-2*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x
-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)
-2*c*x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(
1/2)+4*c*x)*(8*arccosh(c*x)^2+4*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)-1/6912*(-d*(c^2*x^2-1))^(1/2)*(-32*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+32*c^7*x^7+48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-64*c^5*x^5-18*(c*x-1)^(1/2)
*(c*x+1)^(1/2)*c^2*x^2+38*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)*(18*arccosh(c*x)^2+6*arccosh(c*x)+1)*d^2/
(c*x+1)/c^3/(c*x-1)+1/65536*(-d*(c^2*x^2-1))^(1/2)*(-128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+128*c^9*x^9+256*(
c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-320*c^7*x^7-160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+272*c^5*x^5+32*(c*x-1)^
(1/2)*(c*x+1)^(1/2)*c^2*x^2-88*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)*(32*arccosh(c*x)^2+8*arccosh(c*x)+1)
*d^2/(c*x+1)/c^3/(c*x-1))+2*a*b*(-5/256*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*
d^2+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+272*c^5*x^
5-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-88*c^3*x^3+160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(c*x-1)^
(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+8*arccosh(c*x))*d^2/(c*x+1)/c^3/(c*x-1)-1/2304*(-
d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+38*c^3*x^3-48*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*c^4*x^4-6*c*x+18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+6*arccosh
(c*x))*d^2/(c*x+1)/c^3/(c*x-1)+1/1024*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/
2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))*d^2/(c
*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^
(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))*d^2/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2
)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))*d^2/(c*x+1)/c^3/(c*x-1
)+1/1024*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2
)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(1+4*arccosh(c*x))*d^2/(c*x+1)/c^3/(c*x-1)-1/2304*(-d*
(c^2*x^2-1))^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+32*c^7*x^7+48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-
64*c^5*x^5-18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+38*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)*(1+6*arccosh(c
*x))*d^2/(c*x+1)/c^3/(c*x-1)+1/16384*(-d*(c^2*x^2-1))^(1/2)*(-128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+128*c^9*
x^9+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-320*c^7*x^7-160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+272*c^5*x^5+32
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-88*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)*(1+8*arccosh(c*x))*d^2/(c*x
+1)/c^3/(c*x-1))

Fricas [F]

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

[In]

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

[Out]

integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2*x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x
^2)*arccosh(c*x)^2 + 2*(a*b*c^4*d^2*x^6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d),
x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^
2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a^2 + integrate((-c^2*d*x^2 + d)^(5/2)*b^2
*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*x^2*log(c*x + sqrt(c*x + 1)*sqrt(
c*x - 1)), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^2*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^2*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2), x)