\(\int \frac {(\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx\) [76]

Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=-\frac {b c \pi ^{3/2}}{72 x^8}-\frac {5 b c^3 \pi ^{3/2}}{189 x^6}-\frac {b c^5 \pi ^{3/2}}{420 x^4}+\frac {2 b c^7 \pi ^{3/2}}{315 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}+\frac {8}{315} b c^9 \pi ^{3/2} \log (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\frac {\pi ^{3/2} \left (-3675 b c x-7000 b c^3 x^3-630 b c^5 x^5+1680 b c^7 x^7-18264 b c^9 x^9-29400 a \sqrt {1+c^2 x^2}-42000 a c^2 x^2 \sqrt {1+c^2 x^2}-2520 a c^4 x^4 \sqrt {1+c^2 x^2}+3360 a c^6 x^6 \sqrt {1+c^2 x^2}-6720 a c^8 x^8 \sqrt {1+c^2 x^2}-840 b \left (1+c^2 x^2\right )^{5/2} \left (35-20 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+6720 b c^9 x^9 \log (x)\right )}{264600 x^9} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^10,x]
 

Output:

(Pi^(3/2)*(-3675*b*c*x - 7000*b*c^3*x^3 - 630*b*c^5*x^5 + 1680*b*c^7*x^7 - 
 18264*b*c^9*x^9 - 29400*a*Sqrt[1 + c^2*x^2] - 42000*a*c^2*x^2*Sqrt[1 + c^ 
2*x^2] - 2520*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 3360*a*c^6*x^6*Sqrt[1 + c^2*x^ 
2] - 6720*a*c^8*x^8*Sqrt[1 + c^2*x^2] - 840*b*(1 + c^2*x^2)^(5/2)*(35 - 20 
*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] + 6720*b*c^9*x^9*Log[x]))/(264600*x^9)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6219, 27, 1578, 1195, 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.

\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx\)

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int -\frac {\pi \left (c^2 x^2+1\right )^2 \left (8 c^4 x^4-20 c^2 x^2+35\right )}{315 x^9}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{315} \pi ^{3/2} b c \int \frac {\left (c^2 x^2+1\right )^2 \left (8 c^4 x^4-20 c^2 x^2+35\right )}{x^9}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {1}{630} \pi ^{3/2} b c \int \frac {\left (c^2 x^2+1\right )^2 \left (8 c^4 x^4-20 c^2 x^2+35\right )}{x^{10}}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {1}{630} \pi ^{3/2} b c \int \left (\frac {8 c^8}{x^2}-\frac {4 c^6}{x^4}+\frac {3 c^4}{x^6}+\frac {50 c^2}{x^8}+\frac {35}{x^{10}}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{9 \pi x^9}+\frac {4 c^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{63 \pi x^7}-\frac {8 c^4 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{315 \pi x^5}+\frac {1}{630} \pi ^{3/2} b c \left (8 c^8 \log \left (x^2\right )+\frac {4 c^6}{x^2}-\frac {3 c^4}{2 x^4}-\frac {50 c^2}{3 x^6}-\frac {35}{4 x^8}\right )\)

Input:

Int[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^10,x]
 

Output:

-1/9*((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(Pi*x^9) + (4*c^2*(Pi 
+ c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(63*Pi*x^7) - (8*c^4*(Pi + c^2*P 
i*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(315*Pi*x^5) + (b*c*Pi^(3/2)*(-35/(4*x^ 
8) - (50*c^2)/(3*x^6) - (3*c^4)/(2*x^4) + (4*c^6)/x^2 + 8*c^8*Log[x^2]))/6 
30
 

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 1195

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, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[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] && Int 
egerQ[(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. \(2579\) vs. \(2(148)=296\).

Time = 1.22 (sec) , antiderivative size = 2580, normalized size of antiderivative = 14.33

method	result	size
default	\(\text {Expression too large to display}\)	\(2580\)
parts	\(\text {Expression too large to display}\)	\(2580\)

Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))/x^10,x,method=_RETURNVERBOSE)
 

Output:

-16/315*b*Pi^(3/2)*c^9*arcsinh(x*c)+8/315*b*Pi^(3/2)*c^9*ln((x*c+(c^2*x^2+ 
1)^(1/2))^2-1)-64/3*b*Pi^(3/2)/(840*c^12*x^12+945*c^10*x^10+189*c^8*x^8+27 
30*c^6*x^6+6210*c^4*x^4+4725*c^2*x^2+1225)*x^11*(c^2*x^2+1)^(1/2)*arcsinh( 
x*c)*c^20-40/3*b*Pi^(3/2)/(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^2*x^2+1)^(1/2)*arcsinh(x*c)*c 
^18-4/5*b*Pi^(3/2)/(840*c^12*x^12+945*c^10*x^10+189*c^8*x^8+2730*c^6*x^6+6 
210*c^4*x^4+4725*c^2*x^2+1225)*x^7*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^16-313 
9/15*b*Pi^(3/2)/(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^2*x^2+1)^(1/2)*arcsinh(x*c)*c^14+1104/7 
*b*Pi^(3/2)/(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*arcsinh(x*c)*c^13+2785/216*b*Pi^(3/2)/(840*c^1 
2*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+12 
25)*(c^2*x^2+1)*c^9+280/9*b*Pi^(3/2)/(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)*arcsinh(x*c)*c^9-128/315* 
b*Pi^(3/2)/(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^16*c^25+16/105*b*Pi^(3/2)/(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^14*c^23+3 
20/189*b*Pi^(3/2)/(840*c^12*x^12+945*c^10*x^10+189*c^8*x^8+2730*c^6*x^6+62 
10*c^4*x^4+4725*c^2*x^2+1225)*x^12*c^21+8/9*b*Pi^(3/2)/(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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (148) = 296\).

Time = 0.17 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.90 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=-\frac {24 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (8 \, \pi b c^{10} x^{10} + 4 \, \pi b c^{8} x^{8} - \pi b c^{6} x^{6} + 53 \, \pi b c^{4} x^{4} + 85 \, \pi b c^{2} x^{2} + 35 \, \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 96 \, \sqrt {\pi } {\left (\pi b c^{11} x^{11} + \pi b c^{9} x^{9}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (192 \, \pi a c^{10} x^{10} + 96 \, \pi a c^{8} x^{8} - 24 \, \pi a c^{6} x^{6} + 1272 \, \pi a c^{4} x^{4} + 2040 \, \pi a c^{2} x^{2} + 840 \, \pi a - {\left (48 \, \pi b c^{7} x^{7} - 18 \, \pi b c^{5} x^{5} - \pi {\left (48 \, b c^{7} - 18 \, b c^{5} - 200 \, b c^{3} - 105 \, b c\right )} x^{9} - 200 \, \pi b c^{3} x^{3} - 105 \, \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{7560 \, {\left (c^{2} x^{11} + x^{9}\right )}} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^10,x, algorithm="fric 
as")
 

Output:

-1/7560*(24*sqrt(pi + pi*c^2*x^2)*(8*pi*b*c^10*x^10 + 4*pi*b*c^8*x^8 - pi* 
b*c^6*x^6 + 53*pi*b*c^4*x^4 + 85*pi*b*c^2*x^2 + 35*pi*b)*log(c*x + sqrt(c^ 
2*x^2 + 1)) - 96*sqrt(pi)*(pi*b*c^11*x^11 + pi*b*c^9*x^9)*log((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)*(192*pi*a*c^10*x^1 
0 + 96*pi*a*c^8*x^8 - 24*pi*a*c^6*x^6 + 1272*pi*a*c^4*x^4 + 2040*pi*a*c^2* 
x^2 + 840*pi*a - (48*pi*b*c^7*x^7 - 18*pi*b*c^5*x^5 - pi*(48*b*c^7 - 18*b* 
c^5 - 200*b*c^3 - 105*b*c)*x^9 - 200*pi*b*c^3*x^3 - 105*pi*b*c*x)*sqrt(c^2 
*x^2 + 1)))/(c^2*x^11 + x^9)
 

Sympy [F(-1)]

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

integrate((pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x))/x**10,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.13 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\frac {1}{7560} \, {\left (192 \, \pi ^{\frac {3}{2}} c^{8} \log \left (x\right ) + \frac {48 \, \pi ^{\frac {3}{2}} c^{6} x^{6} - 18 \, \pi ^{\frac {3}{2}} c^{4} x^{4} - 200 \, \pi ^{\frac {3}{2}} c^{2} x^{2} - 105 \, \pi ^{\frac {3}{2}}}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{4}}{\pi x^{5}} - \frac {20 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{2}}{\pi x^{7}} + \frac {35 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi x^{9}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{4}}{\pi x^{5}} - \frac {20 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} c^{2}}{\pi x^{7}} + \frac {35 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi x^{9}}\right )} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^10,x, algorithm="maxi 
ma")
 

Output:

1/7560*(192*pi^(3/2)*c^8*log(x) + (48*pi^(3/2)*c^6*x^6 - 18*pi^(3/2)*c^4*x 
^4 - 200*pi^(3/2)*c^2*x^2 - 105*pi^(3/2))/x^8)*b*c - 1/315*b*(8*(pi + pi*c 
^2*x^2)^(5/2)*c^4/(pi*x^5) - 20*(pi + pi*c^2*x^2)^(5/2)*c^2/(pi*x^7) + 35* 
(pi + pi*c^2*x^2)^(5/2)/(pi*x^9))*arcsinh(c*x) - 1/315*a*(8*(pi + pi*c^2*x 
^2)^(5/2)*c^4/(pi*x^5) - 20*(pi + pi*c^2*x^2)^(5/2)*c^2/(pi*x^7) + 35*(pi 
+ pi*c^2*x^2)^(5/2)/(pi*x^9))
 

Giac [F(-2)]

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

integrate((pi*c^2*x^2+pi)^(3/2)*(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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^{10}} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^10,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^10, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^{10}} \, dx=\frac {\sqrt {\pi }\, \pi \left (-8 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}+4 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-50 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-35 \sqrt {c^{2} x^{2}+1}\, a +315 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{10}}d x \right ) b \,x^{9}+315 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{8}}d x \right ) b \,c^{2} x^{9}+8 a \,c^{9} x^{9}\right )}{315 x^{9}} \] Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x))/x^10,x)
 

Output:

(sqrt(pi)*pi*( - 8*sqrt(c**2*x**2 + 1)*a*c**8*x**8 + 4*sqrt(c**2*x**2 + 1) 
*a*c**6*x**6 - 3*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 50*sqrt(c**2*x**2 + 1)* 
a*c**2*x**2 - 35*sqrt(c**2*x**2 + 1)*a + 315*int((sqrt(c**2*x**2 + 1)*asin 
h(c*x))/x**10,x)*b*x**9 + 315*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**8,x) 
*b*c**2*x**9 + 8*a*c**9*x**9))/(315*x**9)