\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx\) [93]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 \sqrt {d-c^2 d x^2}}{28 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 \sqrt {d-c^2 d x^2}}{14 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2} \log (x)}{7 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^7-1/42*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)
^(1/2)+3/28*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/14*b*c^5*d^2*(-c^2*d*x^2+d)^(1/2)
/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*b*c^7*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 = 0.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5917, 74, 272, 45} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^7 d^2 \log (x) \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 d^2 \sqrt {d-c^2 d x^2}}{14 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 \sqrt {d-c^2 d x^2}}{28 x^4 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]

[Out]

-1/42*(b*c*d^2*Sqrt[d - c^2*d*x^2])/(x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(28
*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(14*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(7*Sqrt[-1 +
c*x]*Sqrt[1 + c*x])

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 5917

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 + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), 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*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^3 (1+c x)^3}{x^7} \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^3}{x^7} \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\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 )^3}{x^4} \, dx,x,x^2\right )}{14 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {3 c^2}{x^3}-\frac {3 c^4}{x^2}+\frac {c^6}{x}\right ) \, dx,x,x^2\right )}{14 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 \sqrt {d-c^2 d x^2}}{28 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 \sqrt {d-c^2 d x^2}}{14 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2} \log (x)}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.48 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (12 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))-b c x \left (2-9 c^2 x^2+18 c^4 x^4+12 c^6 x^6 \log (x)\right )\right )}{84 x^7 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) - b*c*x*(2 - 9*c^2*x^2 + 18
*c^4*x^4 + 12*c^6*x^6*Log[x])))/(84*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3774\) vs. \(2(183)=366\).

Time = 1.33 (sec) , antiderivative size = 3775, normalized size of antiderivative = 17.24

method	result	size
default	\(\text {Expression too large to display}\)	\(3775\)
parts	\(\text {Expression too large to display}\)	\(3775\)

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x,method=_RETURNVERBOSE)

[Out]

2/7*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d^2*c^7-1/7*b*(-d*(c^2*x^2-1))^(1/2)/(c*
x-1)^(1/2)/(c*x+1)^(1/2)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*d^2*c^7+55/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7-17/84*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^3/(c*x+
1)/(c*x-1)*c^10+1/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7
*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*c^8+1/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6
*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^7/(c*x+1)/(c*x-1)*arccosh(c*x)-3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c
^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+b*(-d
*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^2/(c*x+1)^(1
/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9+3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*
x^6+21*c^4*x^4-7*c^2*x^2+1)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^17-5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccos
h(c*x)*c^15-b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+
1)*x^12/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^19+5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10
+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-1/7*a/d/x^7*(
-c^2*d*x^2+d)^(7/2)+3/14*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x
^4-7*c^2*x^2+1)*x^13/(c*x+1)/(c*x-1)*c^20-109/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8
*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9-41/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2
/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5+2
3/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^4/
(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-27/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6
*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^11/(c*x+1)/(c*x-1)*c^18+73/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10
*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*c^16-67/42*b*(-d*(c^2*x^2-1))^(1/2)*d^
2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^7/(c*x+1)/(c*x-1)*c^14+11/14*b*(-d
*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5/(c*x+1)/(c
*x-1)*c^12-1/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*
x^2+1)/x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+21/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^
8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^15-119/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/
(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13-1
/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/(c*x+1
)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^7+47/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-
35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11-3/2*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^
12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^17+1/42*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x*c^8+17/
28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5*c^
12-5/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x
^3*c^10-3/14*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2
+1)*x^11*c^18+3/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^
2*x^2+1)*x^9*c^16-83/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^
4-7*c^2*x^2+1)*x^7*c^14+23*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4
*x^4-7*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^16-47*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x
^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14+66*b*(-d*(c^2*x^2-1))^
(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(
c*x)*c^12-66*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2
+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+330/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*
x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-11/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7
*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-165
/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x/(c*x
+1)/(c*x-1)*arccosh(c*x)*c^6+55/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6
+21*c^4*x^4-7*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4+b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^1
0*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^13/(c*x+1)/(c*x-1)*arccosh(c*x)*c^20-7*b*(-d*(c^2*x^2-1
))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^11/(c*x+1)/(c*x-1)*arcc
osh(c*x)*c^18

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.21 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\left [\frac {12 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}, -\frac {12 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 12 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="fricas")

[Out]

[1/84*(12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*l
og(c*x + sqrt(c^2*x^2 - 1)) + 6*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 +
sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) - (18*b*c^5*d^2*x^5 - (18*b*c^
5 - 9*b*c^3 + 2*b*c)*d^2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 12*(a*c
^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7
), -1/84*(12*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*s
qrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^
2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2
*b*c)*d^2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 12*(a*c^8*d^2*x^8 - 4*
a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**8,x)

[Out]

Timed out

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {{\left (6 \, c^{8} d^{4} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + 6 i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{6} d^{\frac {7}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \frac {11 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{4} d^{3}}{x^{2}} - \frac {7 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{2} d^{3}}{x^{4}} + \frac {2 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} d^{3}}{x^{6}}\right )} b c}{84 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arcosh}\left (c x\right )}{7 \, d x^{7}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, d x^{7}} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="maxima")

[Out]

1/84*(6*c^8*d^4*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + 6*I*(-1)^(-2*c^2*d*x^2 + 2*d)*c^6*d^(7/2)*log(-2*c^2*d + 2
*d/x^2) + 11*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*c^4*d^3/x^2 - 7*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*c^2*d^3/x^4
+ 2*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*d^3/x^6)*b*c/d - 1/7*(-c^2*d*x^2 + d)^(7/2)*b*arccosh(c*x)/(d*x^7) - 1
/7*(-c^2*d*x^2 + d)^(7/2)*a/(d*x^7)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^8} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^8,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^8, x)