3.186 \(\int x^3 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=880 \[ -\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^9}{81 \sqrt{c x-1} \sqrt{c x+1}}+\frac{38 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^7}{441 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^6}{3087}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{21 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{9} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{5}{63} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{1}{21} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{484 b^2 d^2 \sqrt{d-c^2 d x^2} x^4}{77175}+\frac{2 b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{189 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^2}{63 c^2}+\frac{3358 b^2 d^2 \sqrt{d-c^2 d x^2} x^2}{694575 c^2}+\frac{4 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 a b d^2 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (c x+1)}-\frac{20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (c x+1)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (c x+1)}-\frac{37384 b^2 d^2 \sqrt{d-c^2 d x^2}}{694575 c^4}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (c x+1)}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (c x+1)} \]

[Out]

(-37384*b^2*d^2*Sqrt[d - c^2*d*x^2])/(694575*c^4) + (3358*b^2*d^2*x^2*Sqrt[d - c^2*d*x^2])/(694575*c^2) + (484
*b^2*d^2*x^4*Sqrt[d - c^2*d*x^2])/77175 - (10*b^2*c^2*d^2*x^6*Sqrt[d - c^2*d*x^2])/3087 + (4*a*b*d^2*x*Sqrt[d
- c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(2835*c^4
*(1 - c*x)*(1 + c*x)) + (8*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(8505*c^4*(1 - c*x)*(1 + c*x)) + (2*b^
2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(4725*c^4*(1 - c*x)*(1 + c*x)) - (20*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d
- c^2*d*x^2])/(3969*c^4*(1 - c*x)*(1 + c*x)) + (2*b^2*d^2*(1 - c^2*x^2)^5*Sqrt[d - c^2*d*x^2])/(729*c^4*(1 -
c*x)*(1 + c*x)) + (4*b^2*d^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*
d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(189*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^5*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (38*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2]*
(a + b*ArcCosh[c*x]))/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh
[c*x]))/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^4) - (d^2
*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^2) + (d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^
2)/21 + (5*d*x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/63 + (x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[
c*x])^2)/9

________________________________________________________________________________________

Rubi [A]  time = 2.34468, antiderivative size = 911, normalized size of antiderivative = 1.04, number of steps used = 34, number of rules used = 18, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.621, Rules used = {5798, 5745, 5743, 5759, 5718, 5654, 74, 5662, 100, 12, 14, 5731, 460, 270, 520, 1251, 897, 1153} \[ -\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^9}{81 \sqrt{c x-1} \sqrt{c x+1}}+\frac{38 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^7}{441 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^6}{3087}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{21 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{21} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{1}{9} d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{5}{63} d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac{484 b^2 d^2 \sqrt{d-c^2 d x^2} x^4}{77175}+\frac{2 b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{189 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^2}{63 c^2}+\frac{3358 b^2 d^2 \sqrt{d-c^2 d x^2} x^2}{694575 c^2}+\frac{4 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 a b d^2 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (c x+1)}-\frac{20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (c x+1)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (c x+1)}-\frac{37384 b^2 d^2 \sqrt{d-c^2 d x^2}}{694575 c^4}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (c x+1)}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (c x+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

(-37384*b^2*d^2*Sqrt[d - c^2*d*x^2])/(694575*c^4) + (3358*b^2*d^2*x^2*Sqrt[d - c^2*d*x^2])/(694575*c^2) + (484
*b^2*d^2*x^4*Sqrt[d - c^2*d*x^2])/77175 - (10*b^2*c^2*d^2*x^6*Sqrt[d - c^2*d*x^2])/3087 + (4*a*b*d^2*x*Sqrt[d
- c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(2835*c^4
*(1 - c*x)*(1 + c*x)) + (8*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(8505*c^4*(1 - c*x)*(1 + c*x)) + (2*b^
2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(4725*c^4*(1 - c*x)*(1 + c*x)) - (20*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d
- c^2*d*x^2])/(3969*c^4*(1 - c*x)*(1 + c*x)) + (2*b^2*d^2*(1 - c^2*x^2)^5*Sqrt[d - c^2*d*x^2])/(729*c^4*(1 -
c*x)*(1 + c*x)) + (4*b^2*d^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*
d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(189*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^5*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (38*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2]*
(a + b*ArcCosh[c*x]))/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh
[c*x]))/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^4) - (d^2
*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^2) + (d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^
2)/21 + (5*d^2*x^4*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/63 + (d^2*x^4*(1 - c*x)^2*(
1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/9

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 5718

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

Rule 5654

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

Rule 74

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

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 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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 1251

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

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^3 \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^3 (-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}{9} d^2 x^4 (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^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (10 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{21 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2835 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (-7+5 c^2 x^2\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{63 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{63 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (-7+5 c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{441 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{105 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{2835 (-1+c x) (1+c x)}\\ &=\frac{2}{525} b^2 d^2 x^4 \sqrt{d-c^2 d x^2}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{525 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{189 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (38 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3087 \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 ) \operatorname{Subst}\left (\int \frac{x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{2835 (-1+c x) (1+c x)}\\ &=-\frac{2 b^2 d^2 x^2 \sqrt{d-c^2 d x^2}}{567 c^2}+\frac{484 b^2 d^2 x^4 \sqrt{d-c^2 d x^2}}{77175}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (38 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15435 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (8 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{525 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{567 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{x^2}{c^2}\right )^2 \left (8-20 x^2+35 x^4\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{2835 (-1+c x) (1+c x)}\\ &=\frac{22 b^2 d^2 x^2 \sqrt{d-c^2 d x^2}}{14175 c^2}+\frac{484 b^2 d^2 x^4 \sqrt{d-c^2 d x^2}}{77175}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (152 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15435 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (8 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1575 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{567 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{63 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^4}-\frac{4 x^2}{c^4}+\frac{3 x^4}{c^4}+\frac{50 x^6}{c^4}+\frac{35 x^8}{c^4}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{2835 (-1+c x) (1+c x)}\\ &=-\frac{40 b^2 d^2 \sqrt{d-c^2 d x^2}}{567 c^4}+\frac{3358 b^2 d^2 x^2 \sqrt{d-c^2 d x^2}}{694575 c^2}+\frac{484 b^2 d^2 x^4 \sqrt{d-c^2 d x^2}}{77175}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac{20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (152 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{46305 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (16 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1575 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{856 b^2 d^2 \sqrt{d-c^2 d x^2}}{14175 c^4}+\frac{3358 b^2 d^2 x^2 \sqrt{d-c^2 d x^2}}{694575 c^2}+\frac{484 b^2 d^2 x^4 \sqrt{d-c^2 d x^2}}{77175}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac{20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (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 (304 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{46305 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{37384 b^2 d^2 \sqrt{d-c^2 d x^2}}{694575 c^4}+\frac{3358 b^2 d^2 x^2 \sqrt{d-c^2 d x^2}}{694575 c^2}+\frac{484 b^2 d^2 x^4 \sqrt{d-c^2 d x^2}}{77175}-\frac{10 b^2 c^2 d^2 x^6 \sqrt{d-c^2 d x^2}}{3087}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac{8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac{20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{9} d^2 x^4 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.707941, size = 288, normalized size = 0.33 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (3969 a^2 \left (7 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^4-126 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (49 c^8 x^8-171 c^6 x^6+189 c^4 x^4-21 c^2 x^2-126\right )+126 b \cosh ^{-1}(c x) \left (63 a \left (7 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^4+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-49 c^8 x^8+171 c^6 x^6-189 c^4 x^4+21 c^2 x^2+126\right )\right )+2 b^2 \left (343 c^{10} x^{10}-1490 c^8 x^8+2152 c^6 x^6-106 c^4 x^4-7039 c^2 x^2+6140\right )+3969 b^2 \left (7 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^4 \cosh ^{-1}(c x)^2\right )}{250047 c^4 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) - 126*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(6140 - 7039*c^2*x^2 - 106*c^4*x^4 + 215
2*c^6*x^6 - 1490*c^8*x^8 + 343*c^10*x^10) + 126*b*(63*a*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) + b*c*x*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]*(126 + 21*c^2*x^2 - 189*c^4*x^4 + 171*c^6*x^6 - 49*c^8*x^8))*ArcCosh[c*x] + 3969*b^2*(-1 + c^2
*x^2)^4*(2 + 7*c^2*x^2)*ArcCosh[c*x]^2))/(250047*c^4*(-1 + c^2*x^2))

________________________________________________________________________________________

Maple [B]  time = 0.536, size = 2224, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)

[Out]

a^2*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2))+b^2*(1/373248*(-d*(c^2*x^2-1))^(1/2)
*(256*x^10*c^10-704*c^8*x^8+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+41*c^2*x^2-120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c
^3+9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(81*arccosh(c*x)^2-18*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/175616
*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x
*c+1)*(49*arccosh(c*x)^2-14*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)+1/1728*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5
*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(9*arccosh(c*x)^2-6*arccos
h(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(ar
ccosh(c*x)^2-2*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1
/2)*x*c+c^2*x^2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)+1/1728*(-d*(c^2*x^2-1))^(1/2)*(-4
*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(9*arccosh(c*x)^
2+6*arccosh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1)-3/175616*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x
^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+1
04*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(49*arccosh(c*x)^2+14*arccosh(c*x)+2)*d^2/(c*x+1)/c
^4/(c*x-1)+1/373248*(-d*(c^2*x^2-1))^(1/2)*(-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+256*x^10*c^10+576*(c*x+1)
^(1/2)*(c*x-1)^(1/2)*x^7*c^7-704*c^8*x^8-432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+688*c^6*x^6+120*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x^3*c^3-280*c^4*x^4-9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+41*c^2*x^2-1)*(81*arccosh(c*x)^2+18*arcco
sh(c*x)+2)*d^2/(c*x+1)/c^4/(c*x-1))+2*a*b*(1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*x^10*c^10-704*c^8*x^8+256*(c*x+
1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x^5*c^5+41*c^2*x^2-120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-
1)*(-1+9*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)-3/25088*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+
1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+7*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)+1/576*
(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x*c+1)*(-1+3*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)
*x*c+c^2*x^2-1)*(-1+arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)
^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)+1/576*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*
(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d^2/(c*x+1)/
c^4/(c*x-1)-3/25088*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)+1/41472*(-d*(c^2*x^2-1))^(1/2)*(-256*(c*x
+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+256*x^10*c^10+576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-704*c^8*x^8-432*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*x^5*c^5+688*c^6*x^6+120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-280*c^4*x^4-9*(c*x+1)^(1/2)*(c
*x-1)^(1/2)*x*c+41*c^2*x^2-1)*(1+9*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1))

________________________________________________________________________________________

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^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.55603, size = 1231, normalized size = 1.4 \begin{align*} \frac{3969 \,{\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 126 \,{\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 126 \,{\left ({\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 63 \,{\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (343 \,{\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \,{\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \,{\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \,{\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} -{\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \,{\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{250047 \,{\left (c^{6} x^{2} - c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

1/250047*(3969*(7*b^2*c^10*d^2*x^10 - 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2*x^6 - 16*b^2*c^4*d^2*x^4 - b^2*c^2*d
^2*x^2 + 2*b^2*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 126*(49*a*b*c^9*d^2*x^9 - 171*a*b*c^
7*d^2*x^7 + 189*a*b*c^5*d^2*x^5 - 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)
- 126*((49*b^2*c^9*d^2*x^9 - 171*b^2*c^7*d^2*x^7 + 189*b^2*c^5*d^2*x^5 - 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x
)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 63*(7*a*b*c^10*d^2*x^10 - 26*a*b*c^8*d^2*x^8 + 34*a*b*c^6*d^2*x^6 -
16*a*b*c^4*d^2*x^4 - a*b*c^2*d^2*x^2 + 2*a*b*d^2)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (343*(
81*a^2 + 2*b^2)*c^10*d^2*x^10 - 2*(51597*a^2 + 1490*b^2)*c^8*d^2*x^8 + 2*(67473*a^2 + 2152*b^2)*c^6*d^2*x^6 -
4*(15876*a^2 + 53*b^2)*c^4*d^2*x^4 - (3969*a^2 + 14078*b^2)*c^2*d^2*x^2 + 2*(3969*a^2 + 6140*b^2)*d^2)*sqrt(-c
^2*d*x^2 + d))/(c^6*x^2 - c^4)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError