\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx\) [78]

Optimal result

Integrand size = 27, antiderivative size = 328 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/9*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^9-4/63*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^7-8/31
5*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5-1/72*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^
(1/2)+5/189*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/420*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^
4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/315*b*c^7*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/315*b*c^9*d
*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 270, 5922, 12, 1265, 907} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^9 d \log (x) \sqrt {d-c^2 d x^2}}{315 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-1/72*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^
6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*
c^7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]
))/(9*d*x^9) - (4*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a
+ b*ArcCosh[c*x]))/(315*d*x^5) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d \left (1-c^2 x^2\right )^2 \left (-35-20 c^2 x^2-8 c^4 x^4\right )}{315 x^9} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-35-20 c^2 x^2-8 c^4 x^4\right )}{x^9} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (-35-20 c^2 x-8 c^4 x^2\right )}{x^5} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {35}{x^5}+\frac {50 c^2}{x^4}-\frac {3 c^4}{x^3}-\frac {4 c^6}{x^2}-\frac {8 c^8}{x}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (840 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+96 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (5+2 c^2 x^2\right ) (a+b \text {arccosh}(c x))+b c x \left (105-200 c^2 x^2+18 c^4 x^4+48 c^6 x^6-192 c^8 x^8 \log (x)\right )\right )}{7560 x^9 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-1/7560*(d*Sqrt[d - c^2*d*x^2]*(840*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 96*c^2*x^2*(-1 + c
*x)^(5/2)*(1 + c*x)^(5/2)*(5 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) + b*c*x*(105 - 200*c^2*x^2 + 18*c^4*x^4 + 48*c^
6*x^6 - 192*c^8*x^8*Log[x])))/(x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4261\) vs. \(2(276)=552\).

Time = 1.29 (sec) , antiderivative size = 4262, normalized size of antiderivative = 12.99

method	result	size
default	\(\text {Expression too large to display}\)	\(4262\)
parts	\(\text {Expression too large to display}\)	\(4262\)

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x,method=_RETURNVERBOSE)

[Out]

-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)*x/(c*x+1)/(c*x-1)*c^10+1225/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^9/(c*x+1)/(c*x-1)*arccosh(c*x)+25915/126*b*(-d*(c^2*x^2-1))^(1/2)*
d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^2/(c*x+1)^(1/2)/(c*x
-1)^(1/2)*c^7-1285/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4
*x^4-4725*c^2*x^2+1225)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5+21175/216*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^1
2-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-1
225/72*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*
x^2+1225)/x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c-1187/60*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+18
9*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13+4189/180*b*(-d*(c^
2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c
*x+1)^(1/2)/(c*x-1)^(1/2)*c^15+280/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*
c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9-829/56*b*(-d*(c^2*x^2-1))
^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c*x+1)^(1/
2)/(c*x-1)^(1/2)*c^11+128/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6
+6210*c^4*x^4-4725*c^2*x^2+1225)*x^17/(c*x+1)/(c*x-1)*c^26-16/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-94
5*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^15/(c*x+1)/(c*x-1)*c^24-344/189*b*(-d*(
c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13
/(c*x+1)/(c*x-1)*c^22-922/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6
+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c*x+1)/(c*x-1)*c^20+2906/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-
945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x-1)*c^18+2069/189*b*(-d
*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^
7/(c*x+1)/(c*x-1)*c^16-4639/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x
^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c*x+1)/(c*x-1)*c^14+455/27*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-9
45*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*c^12+a*(-1/9/d/x^9*(
-c^2*d*x^2+d)^(5/2)+4/9*c^2*(-1/7/d/x^7*(-c^2*d*x^2+d)^(5/2)-2/35*c^2/d/x^5*(-c^2*d*x^2+d)^(5/2)))-16/3*b*(-d*
(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^1
0/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^19+4*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*
c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^17+104/3*b*(-d*(c^2*x^2-1))^(1/2)*d/
(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c*x+1)/(c*x-1)*arc
cosh(c*x)*c^20-16/315*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d*c^9+8/315*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*c^9-43264/63*b*(-d*(c^2
*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x/(c*x+
1)/(c*x-1)*arccosh(c*x)*c^10+113594/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-174520/63*b*(-d*(c^2*x^2-1))^(1/2
)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^3/(c*x+1)/(c*x-1)*
arccosh(c*x)*c^6+1104/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*
c^4*x^4-4725*c^2*x^2+1225)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-120*b*(-d*(c^2*x^2-1))^(1/2)*d/(8
40*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c*x+1)^(1/2)/(c*x-1)^
(1/2)*arccosh(c*x)*c^11+64/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+
6210*c^4*x^4-4725*c^2*x^2+1225)*x^12/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^21-24*b*(-d*(c^2*x^2-1))^(1/2)
*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c*x+1)^(1/2)/(c
*x-1)^(1/2)*arccosh(c*x)*c^19+24/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^
6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^17-208/3*b*(-d*(c^2*x^2-1
))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c*x+1)^(
1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^15+19540/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x
^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-7700/9*b*(-d*(c^2*x^2-1))
^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^7/(c*x+1)/(c*
x-1)*arccosh(c*x)*c^2+59884/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x
^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-212/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(84
0*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x-1)*arccosh
(c*x)*c^18+3151/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x
^4-4725*c^2*x^2+1225)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^16-60632/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^1
2-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14
-64/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)*x^13/(c*x+1)/(c*x-1)*arccosh(c*x)*c^22-30055/504*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x
^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9-40/63*b*(-d*(c^2*
x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7*c^16
-2189/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c
^2*x^2+1225)*x^5*c^14-128/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6
+6210*c^4*x^4-4725*c^2*x^2+1225)*x^15*c^24-16/45*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c
^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13*c^22+1384/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^
12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11*c^20+2306/945*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9*c^18+350/2
7*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1
225)*x^3*c^12-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x
^4-4725*c^2*x^2+1225)*x*c^10

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.20 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\left [-\frac {24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 96 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\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 (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}, \frac {192 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\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 ) - 24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="fricas")

[Out]

[-1/7560*(24*(8*b*c^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-
c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 96*(b*c^11*d*x^11 - b*c^9*d*x^9)*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)) + (48*b*c^7*d*x
^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(
-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*
c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), 1/7560*(192*(b*c^11*d*x^11 - b*c^9*d*x^9)*sqrt(d)*
arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 24*(8*b*c
^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-c^2*d*x^2 + d)*log(
c*x + sqrt(c^2*x^2 - 1)) - (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^
9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^
8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**10,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\frac {1}{7560} \, {\left (192 \, c^{8} \sqrt {-d} d \log \left (x\right ) - \frac {48 \, c^{6} \sqrt {-d} d x^{6} + 18 \, c^{4} \sqrt {-d} d x^{4} - 200 \, c^{2} \sqrt {-d} d x^{2} + 105 \, \sqrt {-d} d}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="maxima")

[Out]

1/7560*(192*c^8*sqrt(-d)*d*log(x) - (48*c^6*sqrt(-d)*d*x^6 + 18*c^4*sqrt(-d)*d*x^4 - 200*c^2*sqrt(-d)*d*x^2 +
105*sqrt(-d)*d)/x^8)*b*c - 1/315*b*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^
7) + 35*(-c^2*d*x^2 + d)^(5/2)/(d*x^9))*arccosh(c*x) - 1/315*a*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^
2*d*x^2 + d)^(5/2)*c^2/(d*x^7) + 35*(-c^2*d*x^2 + d)^(5/2)/(d*x^9))

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{10}} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^10,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^10, x)