\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 302 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=-\frac {b c^3 d^2 \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 \sqrt {d-c^2 d x^2}}{42 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{21 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 (-1+c x)^{7/2} (1+c x)^{7/2} \sqrt {d-c^2 d x^2}}{72 x^8}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}-\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2} \log (x)}{63 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/189*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/42 
*b*c^5*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/21*b*c^7 
*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/72*b*c*d^2*(c* 
x-1)^(7/2)*(c*x+1)^(7/2)*(-c^2*d*x^2+d)^(1/2)/x^8-1/9*(-c^2*d*x^2+d)^(7/2) 
*(a+b*arccosh(c*x))/d/x^9-2/63*c^2*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x)) 
/d/x^7-2/63*b*c^9*d^2*(-c^2*d*x^2+d)^(1/2)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.49 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (168 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+48 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))-b c x \left (21-76 c^2 x^2+90 c^4 x^4-12 c^6 x^6+48 c^8 x^8 \log (x)\right )\right )}{1512 x^9 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^10,x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(168*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcC 
osh[c*x]) + 48*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x 
]) - b*c*x*(21 - 76*c^2*x^2 + 90*c^4*x^4 - 12*c^6*x^6 + 48*c^8*x^8*Log[x]) 
))/(1512*x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.57, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6337, 27, 354, 87, 49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx\)

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (2 c^2 x^2+7\right )}{63 x^9}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (2 c^2 x^2+7\right )}{x^9}dx}{63 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (2 c^2 x^2+7\right )}{x^{10}}dx^2}{126 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \left (2 c^2 \int \frac {\left (1-c^2 x^2\right )^3}{x^8}dx^2-\frac {7 \left (1-c^2 x^2\right )^4}{4 x^8}\right )}{126 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \left (2 c^2 \int \left (-\frac {c^6}{x^2}+\frac {3 c^4}{x^4}-\frac {3 c^2}{x^6}+\frac {1}{x^8}\right )dx^2-\frac {7 \left (1-c^2 x^2\right )^4}{4 x^8}\right )}{126 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{63 d x^7}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (2 c^2 \left (c^6 \left (-\log \left (x^2\right )\right )-\frac {3 c^4}{x^2}+\frac {3 c^2}{2 x^4}-\frac {1}{3 x^6}\right )-\frac {7 \left (1-c^2 x^2\right )^4}{4 x^8}\right )}{126 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^10,x]
 

Output:

-1/9*((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(d*x^9) - (2*c^2*(d - c^ 
2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(63*d*x^7) + (b*c*d^2*Sqrt[d - c^2*d* 
x^2]*((-7*(1 - c^2*x^2)^4)/(4*x^8) + 2*c^2*(-1/3*1/x^6 + (3*c^2)/(2*x^4) - 
 (3*c^4)/x^2 - c^6*Log[x^2])))/(126*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6337

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]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5006\) vs. \(2(254)=508\).

Time = 0.56 (sec) , antiderivative size = 5007, normalized size of antiderivative = 16.58

method	result	size
default	\(\text {Expression too large to display}\)	\(5007\)
parts	\(\text {Expression too large to display}\)	\(5007\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^10,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.64 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\left [\frac {24 \, {\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 24 \, {\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} 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 (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} - {\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 24 \, {\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{1512 \, {\left (c^{2} x^{11} - x^{9}\right )}}, -\frac {48 \, {\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} 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 (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} - {\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 24 \, {\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{1512 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="frica 
s")
 

Output:

[1/1512*(24*(2*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 16*b*c^6*d^2*x^6 + 34*b*c 
^4*d^2*x^4 - 26*b*c^2*d^2*x^2 + 7*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sq 
rt(c^2*x^2 - 1)) + 24*(b*c^11*d^2*x^11 - b*c^9*d^2*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)) + (12*b*c^7*d^2*x^7 - 90*b*c^5*d^2*x^5 - 
 (12*b*c^7 - 90*b*c^5 + 76*b*c^3 - 21*b*c)*d^2*x^9 + 76*b*c^3*d^2*x^3 - 21 
*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 24*(2*a*c^10*d^2*x^10 
 - a*c^8*d^2*x^8 - 16*a*c^6*d^2*x^6 + 34*a*c^4*d^2*x^4 - 26*a*c^2*d^2*x^2 
+ 7*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), -1/1512*(48*(b*c^11*d^2 
*x^11 - b*c^9*d^2*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*(2*b*c^10*d^2 
*x^10 - b*c^8*d^2*x^8 - 16*b*c^6*d^2*x^6 + 34*b*c^4*d^2*x^4 - 26*b*c^2*d^2 
*x^2 + 7*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (12*b* 
c^7*d^2*x^7 - 90*b*c^5*d^2*x^5 - (12*b*c^7 - 90*b*c^5 + 76*b*c^3 - 21*b*c) 
*d^2*x^9 + 76*b*c^3*d^2*x^3 - 21*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2* 
x^2 - 1) - 24*(2*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 16*a*c^6*d^2*x^6 + 34*a 
*c^4*d^2*x^4 - 26*a*c^2*d^2*x^2 + 7*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 
 - x^9)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**10,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.62 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=-\frac {1}{1512} \, {\left (48 \, c^{8} \sqrt {-d} d^{2} \log \left (x\right ) - \frac {12 \, c^{6} \sqrt {-d} d^{2} x^{6} - 90 \, c^{4} \sqrt {-d} d^{2} x^{4} + 76 \, c^{2} \sqrt {-d} d^{2} x^{2} - 21 \, \sqrt {-d} d^{2}}{x^{8}}\right )} b c - \frac {1}{63} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{7}} + \frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{9}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{7}} + \frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{9}}\right )} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="maxim 
a")
 

Output:

-1/1512*(48*c^8*sqrt(-d)*d^2*log(x) - (12*c^6*sqrt(-d)*d^2*x^6 - 90*c^4*sq 
rt(-d)*d^2*x^4 + 76*c^2*sqrt(-d)*d^2*x^2 - 21*sqrt(-d)*d^2)/x^8)*b*c - 1/6 
3*b*(2*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^7) + 7*(-c^2*d*x^2 + d)^(7/2)/(d*x^ 
9))*arccosh(c*x) - 1/63*a*(2*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^7) + 7*(-c^2* 
d*x^2 + d)^(7/2)/(d*x^9))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^{10}} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^10,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^10, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{10}} \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}+\sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-15 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+19 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-7 \sqrt {-c^{2} x^{2}+1}\, a +63 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{10}}d x \right ) b \,x^{9}-126 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{8}}d x \right ) b \,c^{2} x^{9}+63 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{6}}d x \right ) b \,c^{4} x^{9}\right )}{63 x^{9}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))/x^10,x)
 

Output:

(sqrt(d)*d**2*(2*sqrt( - c**2*x**2 + 1)*a*c**8*x**8 + sqrt( - c**2*x**2 + 
1)*a*c**6*x**6 - 15*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 + 19*sqrt( - c**2*x 
**2 + 1)*a*c**2*x**2 - 7*sqrt( - c**2*x**2 + 1)*a + 63*int((sqrt( - c**2*x 
**2 + 1)*acosh(c*x))/x**10,x)*b*x**9 - 126*int((sqrt( - c**2*x**2 + 1)*aco 
sh(c*x))/x**8,x)*b*c**2*x**9 + 63*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/ 
x**6,x)*b*c**4*x**9))/(63*x**9)