\(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [72]

Optimal result

Integrand size = 27, antiderivative size = 281 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-1/16*d*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/8*d*x^3
*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+1/32*b*d*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/96*
b*c*d*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b*c^3*d*x^6*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)-1/32*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5930, 5926, 5939, 5893, 30, 74, 14} \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*d*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*x*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (x^3*(d - c
^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/6 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c
*x]*Sqrt[1 + c*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 74

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

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 5926

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

Rule 5930

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

Rule 5939

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x) (1+c x) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.47 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.96 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-48 a c x \sqrt {d-c^2 d x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )-144 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {18 b \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b \sqrt {d-c^2 d x^2} \left (72 \text {arccosh}(c x)^2-18 \cosh (2 \text {arccosh}(c x))+9 \cosh (4 \text {arccosh}(c x))+2 \cosh (6 \text {arccosh}(c x))-12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{2304 c^3} \]

[In]

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

[Out]

(d*(-48*a*c*x*Sqrt[d - c^2*d*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) - 144*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]
)/(Sqrt[d]*(-1 + c^2*x^2))] - (18*b*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c
*x]*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*(72*ArcCosh[c*x]^2
- 18*Cosh[2*ArcCosh[c*x]] + 9*Cosh[4*ArcCosh[c*x]] + 2*Cosh[6*ArcCosh[c*x]] - 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCo
sh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(2304*c^3
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(882\) vs. \(2(237)=474\).

Time = 0.71 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.14

method	result	size
default	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}+\frac {a d x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{2}}+\frac {a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(883\)
parts	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}+\frac {a d x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{2}}+\frac {a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right ) d}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(883\)

[In]

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

[Out]

-1/6*a*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/24*a/c^2*x*(-c^2*d*x^2+d)^(3/2)+1/16*a/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/16
*a/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-1/32*(-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-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/(c*x+1)/c^3/(c*x-1)+1/512*(-d*(c^2*x^2-1))^(1/2)*(8
*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^
(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))*d/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))*d/(c*x+1)/c^3/(c*x-1)+1/
256*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x
)*(1+2*arccosh(c*x))*d/(c*x+1)/c^3/(c*x-1)+1/512*(-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/(c*x+1)/c^3/(c*x-1)-1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+32*c^7*x^7+4
8*(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/(c*x+1)/c^3/(c*x-1))

Fricas [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

[In]

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

[Out]

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