\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 293 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x-1/3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^3+5/2*c^4*d^
2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-1/6*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1
/4*b*c^5*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/4*c^3*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2
+d)^(1/2)/b/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/3*b*c^3*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5928, 5895, 5893, 30, 74, 14, 272, 45} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c^3 d^2 \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

-1/6*(b*c*d^2*Sqrt[d - c^2*d*x^2])/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2])/(4
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*c^4*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/2 + (5*c^2*d*(d - c^2*
d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - (5*c^3*d^2*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c^3*d^2*Sqrt[d - c^2*d*x^
2]*Log[x])/(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 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 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 5928

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 + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

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

Mathematica [A] (warning: unable to verify)

Time = 1.31 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {30 b c^3 d^3 x^3 (-1+c x) \text {arccosh}(c x)^2-60 a c^3 d^{5/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+3 b c^3 d^3 x^3 (-1+c x) \cosh (2 \text {arccosh}(c x))-4 d^3 \left (b c x (1-c x)+a \sqrt {\frac {-1+c x}{1+c x}} \left (2-16 c^2 x^2+11 c^4 x^4+3 c^6 x^6\right )-14 b c^3 x^3 (-1+c x) \log (c x)\right )-2 b d^3 (-1+c x) \text {arccosh}(c x) \left (4 \sqrt {\frac {-1+c x}{1+c x}} \left (-1-c x+7 c^2 x^2+7 c^3 x^3\right )+3 c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )}{24 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(30*b*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 60*a*c^3*d^(5/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x
^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 3*b*c^3*d^3*x^3*(-1 + c*x)*Cosh[2*ArcCosh[c*x
]] - 4*d^3*(b*c*x*(1 - c*x) + a*Sqrt[(-1 + c*x)/(1 + c*x)]*(2 - 16*c^2*x^2 + 11*c^4*x^4 + 3*c^6*x^6) - 14*b*c^
3*x^3*(-1 + c*x)*Log[c*x]) - 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]*(4*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 - c*x + 7*c^2*x
^2 + 7*c^3*x^3) + 3*c^3*x^3*Sinh[2*ArcCosh[c*x]]))/(24*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\)	\(340\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\)	\(340\)

[In]

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

[Out]

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a*c^
4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/
2)*x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x^3*(-12*(c*x-1)^(1/2)*(c
*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+6*c^5*x^5+30*arccosh(c*x)^2*x^3*c^3+56*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))
^2)*x^3*c^3-56*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-56*c^3*x^3*arccosh(c*x)-3*c^3*x^3+8*arccosh(c*
x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*d^2

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/6*(10*(-c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3*d^(5/2)*arcsin(c*x) + 8*(-
c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d*x^2 + d)^(7/2)/(d*x^3))*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x
+ sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[In]

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

[Out]

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