\(\int x^4 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 454 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/16*d*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/10*x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-3/256*d^2*
x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/128*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*d
^2*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/512*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^
(1/2)+1/512*b*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-31/960*b*c*d^2*x^6*(-c^2*d*x^2+d)^(1/
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+21/640*b*c^3*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/100*b*c
^5*d^2*x^10*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/512*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/
2)/b/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5930, 5926, 5939, 5893, 30, 74, 14, 272, 45} \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{512 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{256 c^4}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(3*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(512*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^4*Sqrt[d - c^2*d*x^2])/(51
2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (31*b*c*d^2*x^6*Sqrt[d - c^2*d*x^2])/(960*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
(21*b*c^3*d^2*x^8*Sqrt[d - c^2*d*x^2])/(640*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^10*Sqrt[d - c^2*d*x^2
])/(100*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(256*c^4) - (d^2*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/32
+ (d*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/16 + (x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/10
- (3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(512*b*c^5*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 45

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

Rule 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 272

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

Rule 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}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 (-1+c x)^2 (1+c x)^2 \, dx}{10 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))+\frac {1}{16} \left (3 d^2\right ) \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 (-1+c x) (1+c x) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right )^2 \, dx}{10 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^2 \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 x^6 \sqrt {d-c^2 d x^2}}{192 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {\left (3 d^2 \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}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (x^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {\left (3 d^2 \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}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))-\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 6.04 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.10 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2880 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )-43200 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+1600 b d^2 \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )+100 b d^2 \sqrt {d-c^2 d x^2} \left (1440 \text {arccosh}(c x)^2-576 \cosh (2 \text {arccosh}(c x))+144 \cosh (4 \text {arccosh}(c x))+64 \cosh (6 \text {arccosh}(c x))+9 \cosh (8 \text {arccosh}(c x))-24 \text {arccosh}(c x) (-48 \sinh (2 \text {arccosh}(c x))+24 \sinh (4 \text {arccosh}(c x))+16 \sinh (6 \text {arccosh}(c x))+3 \sinh (8 \text {arccosh}(c x)))\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-50400 \text {arccosh}(c x)^2+25200 \cosh (2 \text {arccosh}(c x))-3600 \cosh (4 \text {arccosh}(c x))-2600 \cosh (6 \text {arccosh}(c x))-675 \cosh (8 \text {arccosh}(c x))-72 \cosh (10 \text {arccosh}(c x))+120 \text {arccosh}(c x) (-420 \sinh (2 \text {arccosh}(c x))+120 \sinh (4 \text {arccosh}(c x))+130 \sinh (6 \text {arccosh}(c x))+45 \sinh (8 \text {arccosh}(c x))+6 \sinh (10 \text {arccosh}(c x)))\right )}{3686400 c^5 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

Integrate[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(2880*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336
*c^6*x^6 + 128*c^8*x^8) - 43200*a*d^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]
)/(Sqrt[d]*(-1 + c^2*x^2))] + 1600*b*d^2*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*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c
*x]] + Sinh[6*ArcCosh[c*x]])) + 100*b*d^2*Sqrt[d - c^2*d*x^2]*(1440*ArcCosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]]
+ 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] - 24*ArcCosh[c*x]*(-48*Sinh[2*Ar
cCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCosh[c*x]])) + b*d^2*Sqrt[d - c^
2*d*x^2]*(-50400*ArcCosh[c*x]^2 + 25200*Cosh[2*ArcCosh[c*x]] - 3600*Cosh[4*ArcCosh[c*x]] - 2600*Cosh[6*ArcCosh
[c*x]] - 675*Cosh[8*ArcCosh[c*x]] - 72*Cosh[10*ArcCosh[c*x]] + 120*ArcCosh[c*x]*(-420*Sinh[2*ArcCosh[c*x]] + 1
20*Sinh[4*ArcCosh[c*x]] + 130*Sinh[6*ArcCosh[c*x]] + 45*Sinh[8*ArcCosh[c*x]] + 6*Sinh[10*ArcCosh[c*x]])))/(368
6400*c^5*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. \(1742\) vs. \(2(386)=772\).

Time = 0.90 (sec) , antiderivative size = 1743, normalized size of antiderivative = 3.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1743\)
parts	\(\text {Expression too large to display}\)	\(1743\)

[In]

int(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

-1/10*a*x^3*(-c^2*d*x^2+d)^(7/2)/c^2/d-3/80*a/c^4*x*(-c^2*d*x^2+d)^(7/2)/d+1/160*a/c^4*x*(-c^2*d*x^2+d)^(5/2)+
1/128*a/c^4*d*x*(-c^2*d*x^2+d)^(3/2)+3/256*a/c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+3/256*a/c^4*d^3/(c^2*d)^(1/2)*arct
an((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/512*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^5*arcc
osh(c*x)^2*d^2+1/102400*(-d*(c^2*x^2-1))^(1/2)*(-1280*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+512*(c*x+1)^(1/2)*(c
*x-1)^(1/2)*c^10*x^10+50*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2)-832*c^5*x^5+170*c^3*x
^3+1696*c^7*x^7+1120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-400*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-1536*c^9*x^9-
10*c*x+512*c^11*x^11)*(-1+10*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)-1/32768*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9
-320*c^7*x^7+128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+272*c^5*x^5-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6-88*c^
3*x^3+160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c*x-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+
1)^(1/2))*(-1+8*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)-1/12288*(-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^5/(c*x-1)+1/2048*(-d*(c^2
*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)
*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)+1/2048*(-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^5/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1
/2)*(c*x+1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^
(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(
1/2)+4*c*x)*(1+4*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)-1/12288*(-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^5/(c*x-1)-1/32768*(-d*(c
^2*x^2-1))^(1/2)*(-128*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8+128*c^9*x^9+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6
-320*c^7*x^7-160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+272*c^5*x^5+32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-88*c^3
*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)*(1+8*arccosh(c*x))*d^2/(c*x+1)/c^5/(c*x-1)+1/102400*(-d*(c^2*x^2-1))^(
1/2)*(-512*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^10*x^10+512*c^11*x^11+1280*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^8*x^8-1536*c
^9*x^9-1120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^6*x^6+1696*c^7*x^7+400*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4-832*c^5*x
^5-50*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+170*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-10*c*x)*(1+10*arccosh(c*x))*
d^2/(c*x+1)/c^5/(c*x-1))

Fricas [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^8 - 2*a*c^2*d^2*x^6 + a*d^2*x^4 + (b*c^4*d^2*x^8 - 2*b*c^2*d^2*x^6 + b*d^2*x^4)*arccosh(
c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F(-1)]

Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**4*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/1280*(128*(-c^2*d*x^2 + d)^(7/2)*x^3/(c^2*d) - 8*(-c^2*d*x^2 + d)^(5/2)*x/c^4 + 48*(-c^2*d*x^2 + d)^(7/2)*x
/(c^4*d) - 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^4 - 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^4 - 15*d^(5/2)*arcsin(c*x)/c^5)
*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

Giac [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)*x^4, x)

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[In]

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), x)