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

Optimal. Leaf size=841 \[ -\frac{b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^8}{32 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^4 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^7}{256 (1-c x) (c x+1)}+\frac{17 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^6}{144 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{864} b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^5-\frac{43 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^5}{4608 (1-c x) (c x+1)}-\frac{59 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^4}{384 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{8} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{5}{48} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{5}{64} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{215 b^2 d^2 \sqrt{d-c^2 d x^2} x^3}{13824}+\frac{73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^3}{18432 (1-c x) (c x+1)}+\frac{5 b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{128 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x}{128 c^2}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} x}{9216 c^2}+\frac{73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x}{12288 c^2 (1-c x) (c x+1)}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{73 b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{12288 c^3 (1-c x) (c x+1)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 2.12388, antiderivative size = 872, normalized size of antiderivative = 1.04, number of steps used = 30, number of rules used = 21, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.724, Rules used = {5798, 5745, 5743, 5759, 5676, 5662, 90, 52, 100, 12, 14, 5731, 460, 266, 43, 520, 1267, 459, 321, 217, 206} \[ -\frac{b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^8}{32 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^4 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^7}{256 (1-c x) (c x+1)}+\frac{17 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^6}{144 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{864} b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^5-\frac{43 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^5}{4608 (1-c x) (c x+1)}-\frac{59 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^4}{384 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{64} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{1}{8} d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{5}{48} d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac{215 b^2 d^2 \sqrt{d-c^2 d x^2} x^3}{13824}+\frac{73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x^3}{18432 (1-c x) (c x+1)}+\frac{5 b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{128 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x}{128 c^2}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} x}{9216 c^2}+\frac{73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x}{12288 c^2 (1-c x) (c x+1)}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{73 b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{12288 c^3 (1-c x) (c x+1)} \]

Antiderivative was successfully verified.

[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^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d -
c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/48 + (d^2*x^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*
x])^2)/8 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[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 5798

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

Rule 5745

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

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[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, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

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 5676

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

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 90

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 52

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 100

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 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 5731

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 460

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 266

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 43

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 520

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 1267

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 459

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 321

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 217

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 206

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

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{8 \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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{12 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{16 \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 ) \left (a+b \cosh ^{-1}(c x)\right ) \, 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{11 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{32 \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 \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{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{64 c \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 (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 (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}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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}{\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}}{256 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 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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{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 (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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 (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 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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 ) \operatorname{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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)}\\ \end{align*}

Mathematica [A]  time = 5.82776, size = 910, normalized size = 1.08 \[ -\frac{d^2 \left (-110592 a^2 c^8 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^8-110592 a^2 c^7 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^7+313344 a^2 c^6 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^6+313344 a^2 c^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^5-271872 a^2 c^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^4-271872 a^2 c^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^3+34560 a^2 c^2 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^2+34560 a^2 c \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right ) x+34560 a^2 c \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x+11520 b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^3+34560 a^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+13824 a b \sqrt{d-c^2 d x^2} \cosh \left (2 \cosh ^{-1}(c x)\right )+3456 a b \sqrt{d-c^2 d x^2} \cosh \left (4 \cosh ^{-1}(c x)\right )-1536 a b \sqrt{d-c^2 d x^2} \cosh \left (6 \cosh ^{-1}(c x)\right )+216 a b \sqrt{d-c^2 d x^2} \cosh \left (8 \cosh ^{-1}(c x)\right )-6912 b^2 \sqrt{d-c^2 d x^2} \sinh \left (2 \cosh ^{-1}(c x)\right )-864 b^2 \sqrt{d-c^2 d x^2} \sinh \left (4 \cosh ^{-1}(c x)\right )+256 b^2 \sqrt{d-c^2 d x^2} \sinh \left (6 \cosh ^{-1}(c x)\right )-27 b^2 \sqrt{d-c^2 d x^2} \sinh \left (8 \cosh ^{-1}(c x)\right )+24 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \left (576 b \cosh \left (2 \cosh ^{-1}(c x)\right )+144 b \cosh \left (4 \cosh ^{-1}(c x)\right )-64 b \cosh \left (6 \cosh ^{-1}(c x)\right )+9 b \cosh \left (8 \cosh ^{-1}(c x)\right )-1152 a \sinh \left (2 \cosh ^{-1}(c x)\right )-576 a \sinh \left (4 \cosh ^{-1}(c x)\right )+384 a \sinh \left (6 \cosh ^{-1}(c x)\right )-72 a \sinh \left (8 \cosh ^{-1}(c x)\right )\right )-288 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left (-120 a+48 b \sinh \left (2 \cosh ^{-1}(c x)\right )+24 b \sinh \left (4 \cosh ^{-1}(c x)\right )-16 b \sinh \left (6 \cosh ^{-1}(c x)\right )+3 b \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{884736 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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)]*Sqrt[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*Sqrt[(-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*Sqrt[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*Sqrt[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*Sqrt[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*ArcCos
h[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*ArcCosh[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*ArcCosh[c*x]]))
)/(884736*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

________________________________________________________________________________________

Maple [A]  time = 0.533, size = 1312, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6/(c*x-1)*arccosh(c*x)*x^9-23
/24*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^7+127/96*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5+5/64*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x-5/
128*a*b*(-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^2-133/192*a*b*(-d*(c^2*x^2-1))
^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+17/144*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1
/2)*arccosh(c*x)*x^6+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6/(c*x-1)*arccosh(c*x)^2*x^9-23/48*b^2*(-d*(
c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)^2*x^7+127/192*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^
2/(c*x-1)*arccosh(c*x)^2*x^5+5/128*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)^2*x-59/384*
b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*arccosh(c*x)*x^4+5/128*b^2*(-d*(c^2*x^2-1))^(1/2)
*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*arccosh(c*x)*x^2-1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^5/(c*x
-1)^(1/2)*arccosh(c*x)*x^8-1/32*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^5/(c*x-1)^(1/2)*x^8+17/144*a*b*
(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^6-59/384*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(
1/2)*c/(c*x-1)^(1/2)*x^4+5/128*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^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))-1081/1
10592*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*x^3+5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)-1/8*a^2*x*(-c^
2*d*x^2+d)^(7/2)/c^2/d+359/36864*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)*arccosh(c*x)+3
59/36864*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)-5/384*b^2*(-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+1/256*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6/(c*x-1)*x^9-263
/13824*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^4/(c*x-1)*x^7+1915/55296*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1
)*c^2/(c*x-1)*x^5-359/36864*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/c^2/(c*x-1)*x-133/384*b^2*(-d*(c^2*x^2-1))^
(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x^3

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{6} - 2 \, a^{2} c^{2} d^{2} x^{4} + a^{2} d^{2} x^{2} +{\left (b^{2} c^{4} d^{2} x^{6} - 2 \, b^{2} c^{2} d^{2} x^{4} + b^{2} d^{2} x^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{6} - 2 \, a b c^{2} d^{2} x^{4} + a b d^{2} x^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out