\(\int (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) [189]

Optimal result

Integrand size = 26, antiderivative size = 486 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

5/24*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/6*x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2+245/1152*b^
2*d^2*x*(-c^2*d*x^2+d)^(1/2)+65/1728*b^2*d^2*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)+1/108*b^2*d^2*x*(-c*x+1)^
2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)+5/16*d^2*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+115/1152*b^2*d^2*arccosh
(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/16*b*c*d^2*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1
/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/48*b*d^2*(-c^2*x^2+1)^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)+1/18*b*d^2*(-c^2*x^2+1)^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1
/2)-5/48*d^2*(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5897, 5895, 5893, 5883, 92, 54, 5912, 5914, 38} \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {115 b^2 d^2 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{1152 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2}+\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{1728} \]

[In]

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

[Out]

(245*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/1728 + (b^2*
d^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2])/108 + (115*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (5*b*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(48*c*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(18*c*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)/16 + (5*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x
])^2)/24 + (x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/6 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]
)^3)/(48*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 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 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 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

Rule 5897

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

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 5914

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} (5 d) \int \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 (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} \left (5 d^2\right ) \int \sqrt {d-c^2 d x^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 \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{3 \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 (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{12 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \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}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{18 \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 \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{12 \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 (a+b \text {arccosh}(c x)) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \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}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{108 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{48 \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^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{32} b^2 d^2 x \sqrt {d-c^2 d x^2}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \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}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{144 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{64 \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}{32 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \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}{48 b c \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}{288 \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}{128 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \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}{48 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.65 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.52 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \left (9504 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+9504 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-1440 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3-4320 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-4320 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-3240 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+324 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-24 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+1620 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-81 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+4 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-12 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (270 b \cosh (2 \text {arccosh}(c x))-27 b \cosh (4 \text {arccosh}(c x))+2 b \cosh (6 \text {arccosh}(c x))-540 a \sinh (2 \text {arccosh}(c x))+108 a \sinh (4 \text {arccosh}(c x))-12 a \sinh (6 \text {arccosh}(c x)))+72 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-60 a+45 b \sinh (2 \text {arccosh}(c x))-9 b \sinh (4 \text {arccosh}(c x))+b \sinh (6 \text {arccosh}(c x)))\right )}{13824 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

(d^2*(9504*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 9504*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)
]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^4*x^4*Sqr
t[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]
+ 2304*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 1440*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^
3 - 4320*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))] - 4
320*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))] - 32
40*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 324*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 24*a*b*Sq
rt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 1620*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 81*b^2*Sqrt[d - c
^2*d*x^2]*Sinh[4*ArcCosh[c*x]] + 4*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 12*b*Sqrt[d - c^2*d*x^2]*Arc
Cosh[c*x]*(270*b*Cosh[2*ArcCosh[c*x]] - 27*b*Cosh[4*ArcCosh[c*x]] + 2*b*Cosh[6*ArcCosh[c*x]] - 540*a*Sinh[2*Ar
cCosh[c*x]] + 108*a*Sinh[4*ArcCosh[c*x]] - 12*a*Sinh[6*ArcCosh[c*x]]) + 72*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^
2*(-60*a + 45*b*Sinh[2*ArcCosh[c*x]] - 9*b*Sinh[4*ArcCosh[c*x]] + b*Sinh[6*ArcCosh[c*x]])))/(13824*c*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. \(1736\) vs. \(2(422)=844\).

Time = 1.05 (sec) , antiderivative size = 1737, normalized size of antiderivative = 3.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1737\)
parts	\(\text {Expression too large to display}\)	\(1737\)

[In]

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

[Out]

1/6*x*(-c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a^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*(-5/48*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)/c*arccosh(c*x)^3*d^2+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*x+1)/c-3/1024*(-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))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)*d^2/(c*x-1)/(c*x+1)/c+15/256*(-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*x+1)/c+15/256*(-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*x+1)/c-3/1
024*(-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*x+1)/c+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*x+1)/c)+2*a*b*(-5/32*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x
+1)^(1/2)/c*arccosh(c*x)^2*d^2+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*x+1)/c-3/512*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-1
2*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*x+1)/c+15/256*(-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*x+1)/c+15/256*(-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*a
rccosh(c*x))*d^2/(c*x-1)/(c*x+1)/c-3/512*(-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*x+1)/c+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*x+1)/c)

Fricas [F]

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

[In]

integrate((-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^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcc
osh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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