\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx\) [88]

Optimal result

Integrand size = 26, antiderivative size = 156 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\frac {b c^3 \pi ^{5/2}}{189 x^6}+\frac {b c^5 \pi ^{5/2}}{42 x^4}+\frac {b c^7 \pi ^{5/2}}{21 x^2}-\frac {b c \pi ^{5/2} \left (1+c^2 x^2\right )^4}{72 x^8}-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {2}{63} b c^9 \pi ^{5/2} \log (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=-\frac {\pi ^{5/2} \left (735 b c x+2660 b c^3 x^3+3150 b c^5 x^5+420 b c^7 x^7-4566 b c^9 x^9+5880 a \sqrt {1+c^2 x^2}+15960 a c^2 x^2 \sqrt {1+c^2 x^2}+12600 a c^4 x^4 \sqrt {1+c^2 x^2}+840 a c^6 x^6 \sqrt {1+c^2 x^2}-1680 a c^8 x^8 \sqrt {1+c^2 x^2}-840 b \left (1+c^2 x^2\right )^{7/2} \left (-7+2 c^2 x^2\right ) \text {arcsinh}(c x)+1680 b c^9 x^9 \log (x)\right )}{52920 x^9} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^10,x]
 

Output:

-1/52920*(Pi^(5/2)*(735*b*c*x + 2660*b*c^3*x^3 + 3150*b*c^5*x^5 + 420*b*c^ 
7*x^7 - 4566*b*c^9*x^9 + 5880*a*Sqrt[1 + c^2*x^2] + 15960*a*c^2*x^2*Sqrt[1 
 + c^2*x^2] + 12600*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 840*a*c^6*x^6*Sqrt[1 + c 
^2*x^2] - 1680*a*c^8*x^8*Sqrt[1 + c^2*x^2] - 840*b*(1 + c^2*x^2)^(7/2)*(-7 
 + 2*c^2*x^2)*ArcSinh[c*x] + 1680*b*c^9*x^9*Log[x]))/x^9
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6219, 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.

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

\(\Big \downarrow \) 6219

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \pi ^{5/2} b c \int \frac {\left (7-2 c^2 x^2\right ) \left (c^2 x^2+1\right )^3}{x^9}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{126} \pi ^{5/2} b c \int \frac {\left (7-2 c^2 x^2\right ) \left (c^2 x^2+1\right )^3}{x^{10}}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{126} \pi ^{5/2} b c \left (-2 c^2 \int \frac {\left (c^2 x^2+1\right )^3}{x^8}dx^2-\frac {7 \left (c^2 x^2+1\right )^4}{4 x^8}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{126} \pi ^{5/2} b c \left (-2 c^2 \int \left (\frac {c^6}{x^2}+\frac {3 c^4}{x^4}+\frac {3 c^2}{x^6}+\frac {1}{x^8}\right )dx^2-\frac {7 \left (c^2 x^2+1\right )^4}{4 x^8}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {2 c^2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}+\frac {1}{126} \pi ^{5/2} b c \left (-\frac {7 \left (c^2 x^2+1\right )^4}{4 x^8}-2 c^2 \left (c^6 \log \left (x^2\right )-\frac {3 c^4}{x^2}-\frac {3 c^2}{2 x^4}-\frac {1}{3 x^6}\right )\right )\)

Input:

Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^10,x]
 

Output:

-1/9*((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(Pi*x^9) + (2*c^2*(Pi 
+ c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(63*Pi*x^7) + (b*c*Pi^(5/2)*((-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
 

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 6219

Int[((a_.) + ArcSinh[(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*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && 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. \(2988\) vs. \(2(128)=256\).

Time = 1.40 (sec) , antiderivative size = 2989, normalized size of antiderivative = 19.16

method	result	size
default	\(\text {Expression too large to display}\)	\(2989\)
parts	\(\text {Expression too large to display}\)	\(2989\)

Input:

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))/x^10,x,method=_RETURNVERBOSE)
 

Output:

-1/18*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-115 
5*c^6*x^6-837*c^4*x^4-315*c^2*x^2-49)*x^10*c^19+14843/216*b*Pi^(5/2)/(63*c 
^14*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-837*c^4*x^4- 
315*c^2*x^2-49)*(c^2*x^2+1)*c^9+14/9*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x^1 
2-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-837*c^4*x^4-315*c^2*x^2-49)*arcsi 
nh(x*c)*c^9-2/63*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819* 
c^8*x^8-1155*c^6*x^6-837*c^4*x^4-315*c^2*x^2-49)*x^16*c^25-5/21*b*Pi^(5/2) 
/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-837*c^ 
4*x^4-315*c^2*x^2-49)*x^14*c^23-38/189*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x 
^12-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-837*c^4*x^4-315*c^2*x^2-49)*x^1 
2*c^21+a*(-1/9/Pi/x^9*(Pi*c^2*x^2+Pi)^(7/2)+2/63/Pi*c^2/x^7*(Pi*c^2*x^2+Pi 
)^(7/2))+665/108*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819* 
c^8*x^8-1155*c^6*x^6-837*c^4*x^4-315*c^2*x^2-49)/x^6*(c^2*x^2+1)*c^3+49/72 
*b*Pi^(5/2)/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-1155*c^6 
*x^6-837*c^4*x^4-315*c^2*x^2-49)/x^8*(c^2*x^2+1)*c+49/9*b*Pi^(5/2)/(63*c^1 
4*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-837*c^4*x^4-31 
5*c^2*x^2-49)/x^9*(c^2*x^2+1)^(1/2)*arcsinh(x*c)+4/63*b*Pi^(5/2)*c^9*arcsi 
nh(x*c)-2/63*b*Pi^(5/2)*c^9*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)-3041/756*b*Pi^ 
(5/2)/(63*c^14*x^14+105*c^12*x^12-189*c^10*x^10-819*c^8*x^8-1155*c^6*x^6-8 
37*c^4*x^4-315*c^2*x^2-49)*x^8*(c^2*x^2+1)*c^17+26*b*Pi^(5/2)/(63*c^14*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (128) = 256\).

Time = 0.14 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.44 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\frac {24 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, \pi ^{2} b c^{10} x^{10} + \pi ^{2} b c^{8} x^{8} - 16 \, \pi ^{2} b c^{6} x^{6} - 34 \, \pi ^{2} b c^{4} x^{4} - 26 \, \pi ^{2} b c^{2} x^{2} - 7 \, \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 24 \, \sqrt {\pi } {\left (\pi ^{2} b c^{11} x^{11} + \pi ^{2} b c^{9} x^{9}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} - \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (48 \, \pi ^{2} a c^{10} x^{10} + 24 \, \pi ^{2} a c^{8} x^{8} - 384 \, \pi ^{2} a c^{6} x^{6} - 816 \, \pi ^{2} a c^{4} x^{4} - 624 \, \pi ^{2} a c^{2} x^{2} - 168 \, \pi ^{2} a - {\left (12 \, \pi ^{2} b c^{7} x^{7} + 90 \, \pi ^{2} b c^{5} x^{5} - \pi ^{2} {\left (12 \, b c^{7} + 90 \, b c^{5} + 76 \, b c^{3} + 21 \, b c\right )} x^{9} + 76 \, \pi ^{2} b c^{3} x^{3} + 21 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{1512 \, {\left (c^{2} x^{11} + x^{9}\right )}} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^10,x, algorithm="fric 
as")
 

Output:

1/1512*(24*sqrt(pi + pi*c^2*x^2)*(2*pi^2*b*c^10*x^10 + pi^2*b*c^8*x^8 - 16 
*pi^2*b*c^6*x^6 - 34*pi^2*b*c^4*x^4 - 26*pi^2*b*c^2*x^2 - 7*pi^2*b)*log(c* 
x + sqrt(c^2*x^2 + 1)) + 24*sqrt(pi)*(pi^2*b*c^11*x^11 + pi^2*b*c^9*x^9)*l 
og((pi + pi*c^2*x^6 + pi*c^2*x^2 + pi*x^4 - sqrt(pi)*sqrt(pi + pi*c^2*x^2) 
*sqrt(c^2*x^2 + 1)*(x^4 - 1))/(c^2*x^4 + x^2)) + sqrt(pi + pi*c^2*x^2)*(48 
*pi^2*a*c^10*x^10 + 24*pi^2*a*c^8*x^8 - 384*pi^2*a*c^6*x^6 - 816*pi^2*a*c^ 
4*x^4 - 624*pi^2*a*c^2*x^2 - 168*pi^2*a - (12*pi^2*b*c^7*x^7 + 90*pi^2*b*c 
^5*x^5 - pi^2*(12*b*c^7 + 90*b*c^5 + 76*b*c^3 + 21*b*c)*x^9 + 76*pi^2*b*c^ 
3*x^3 + 21*pi^2*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^2*x^11 + x^9)
 

Sympy [F(-1)]

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

integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**10,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=-\frac {1}{1512} \, {\left (48 \, \pi ^{\frac {5}{2}} c^{8} \log \left (x\right ) + \frac {12 \, \pi ^{\frac {5}{2}} c^{6} x^{6} + 90 \, \pi ^{\frac {5}{2}} c^{4} x^{4} + 76 \, \pi ^{\frac {5}{2}} c^{2} x^{2} + 21 \, \pi ^{\frac {5}{2}}}{x^{8}}\right )} b c + \frac {1}{63} \, b {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{2}}{\pi x^{7}} - \frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi x^{9}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, a {\left (\frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} c^{2}}{\pi x^{7}} - \frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi x^{9}}\right )} \] Input:

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^10,x, algorithm="maxi 
ma")
 

Output:

-1/1512*(48*pi^(5/2)*c^8*log(x) + (12*pi^(5/2)*c^6*x^6 + 90*pi^(5/2)*c^4*x 
^4 + 76*pi^(5/2)*c^2*x^2 + 21*pi^(5/2))/x^8)*b*c + 1/63*b*(2*(pi + pi*c^2* 
x^2)^(7/2)*c^2/(pi*x^7) - 7*(pi + pi*c^2*x^2)^(7/2)/(pi*x^9))*arcsinh(c*x) 
 + 1/63*a*(2*(pi + pi*c^2*x^2)^(7/2)*c^2/(pi*x^7) - 7*(pi + pi*c^2*x^2)^(7 
/2)/(pi*x^9))
 

Giac [F(-2)]

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

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^{10}} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^10,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^10, x)
 

Reduce [F]

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

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x))/x^10,x)
 

Output:

(sqrt(pi)*pi**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)*asinh(c 
*x))/x**10,x)*b*x**9 + 126*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**8,x)*b* 
c**2*x**9 + 63*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**6,x)*b*c**4*x**9 - 
2*a*c**9*x**9))/(63*x**9)