Integrand size = 26, antiderivative size = 141 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b \pi ^{5/2} x}{63 c^3}-\frac {b \pi ^{5/2} x^3}{189 c}-\frac {1}{21} b c \pi ^{5/2} x^5-\frac {19}{441} b c^3 \pi ^{5/2} x^7-\frac {1}{81} b c^5 \pi ^{5/2} x^9-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2} \] Output:
2/63*b*Pi^(5/2)*x/c^3-1/189*b*Pi^(5/2)*x^3/c-1/21*b*c*Pi^(5/2)*x^5-19/441* b*c^3*Pi^(5/2)*x^7-1/81*b*c^5*Pi^(5/2)*x^9-1/7*(Pi*c^2*x^2+Pi)^(7/2)*(a+b* arcsinh(c*x))/c^4/Pi+1/9*(Pi*c^2*x^2+Pi)^(9/2)*(a+b*arcsinh(c*x))/c^4/Pi^2
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.77 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{5/2} \left (63 a \left (1+c^2 x^2\right )^{7/2} \left (-2+7 c^2 x^2\right )-b c x \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+63 b \left (1+c^2 x^2\right )^{7/2} \left (-2+7 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3969 c^4} \] Input:
Integrate[x^3*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(Pi^(5/2)*(63*a*(1 + c^2*x^2)^(7/2)*(-2 + 7*c^2*x^2) - b*c*x*(-126 + 21*c^ 2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 63*b*(1 + c^2*x^2)^(7/2) *(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4)
Time = 0.44 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 290, 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.
| \(\displaystyle \int x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {\pi ^2 \left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3}{63 c^4}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\pi ^{5/2} b \int \left (2-7 c^2 x^2\right ) \left (c^2 x^2+1\right )^3dx}{63 c^3}+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^4}\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {\pi ^{5/2} b \int \left (-7 c^8 x^8-19 c^6 x^6-15 c^4 x^4-c^2 x^2+2\right )dx}{63 c^3}+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^4}+\frac {\pi ^{5/2} b \left (-\frac {7}{9} c^8 x^9-\frac {19 c^6 x^7}{7}-3 c^4 x^5-\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\) |
Input:
Int[x^3*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
Output:
(b*Pi^(5/2)*(2*x - (c^2*x^3)/3 - 3*c^4*x^5 - (19*c^6*x^7)/7 - (7*c^8*x^9)/ 9))/(63*c^3) - ((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*Pi) + ((Pi + c^2*Pi*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*Pi^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
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])
Time = 1.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.54
| method | result | size |
| orering | \(\frac {\left (833 c^{10} x^{10}+3153 c^{8} x^{8}+4167 c^{6} x^{6}+1743 c^{4} x^{4}-1008 c^{2} x^{2}-504\right ) \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{3969 c^{4} \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (49 c^{8} x^{8}+171 c^{6} x^{6}+189 c^{4} x^{4}+21 c^{2} x^{2}-126\right ) \left (3 x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+5 x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}+\frac {x^{3} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3969 x^{2} c^{4} \left (c^{2} x^{2}+1\right )^{2}}\) | \(217\) |
| default | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (441 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}+1638 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-49 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+2142 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-171 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+1008 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-189 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}-63 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-21 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-126 \,\operatorname {arcsinh}\left (x c \right )+126 \sqrt {c^{2} x^{2}+1}\, x c \right )}{3969 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(226\) |
| parts | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (441 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}+1638 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-49 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+2142 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-171 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}+1008 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-189 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}-63 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-21 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-126 \,\operatorname {arcsinh}\left (x c \right )+126 \sqrt {c^{2} x^{2}+1}\, x c \right )}{3969 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(226\) |
Input:
int(x^3*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/3969*(833*c^10*x^10+3153*c^8*x^8+4167*c^6*x^6+1743*c^4*x^4-1008*c^2*x^2- 504)/c^4/(c^2*x^2+1)^3*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))-1/3969/x^2 *(49*c^8*x^8+171*c^6*x^6+189*c^4*x^4+21*c^2*x^2-126)/c^4/(c^2*x^2+1)^2*(3* x^2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c))+5*x^4*(Pi*c^2*x^2+Pi)^(3/2)*( a+b*arcsinh(x*c))*Pi*c^2+x^3*(Pi*c^2*x^2+Pi)^(5/2)*b*c/(c^2*x^2+1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (113) = 226\).
Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.87 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {63 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (7 \, \pi ^{2} b c^{10} x^{10} + 26 \, \pi ^{2} b c^{8} x^{8} + 34 \, \pi ^{2} b c^{6} x^{6} + 16 \, \pi ^{2} b c^{4} x^{4} - \pi ^{2} b c^{2} x^{2} - 2 \, \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (441 \, \pi ^{2} a c^{10} x^{10} + 1638 \, \pi ^{2} a c^{8} x^{8} + 2142 \, \pi ^{2} a c^{6} x^{6} + 1008 \, \pi ^{2} a c^{4} x^{4} - 63 \, \pi ^{2} a c^{2} x^{2} - 126 \, \pi ^{2} a - {\left (49 \, \pi ^{2} b c^{9} x^{9} + 171 \, \pi ^{2} b c^{7} x^{7} + 189 \, \pi ^{2} b c^{5} x^{5} + 21 \, \pi ^{2} b c^{3} x^{3} - 126 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{3969 \, {\left (c^{6} x^{2} + c^{4}\right )}} \] Input:
integrate(x^3*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="frica s")
Output:
1/3969*(63*sqrt(pi + pi*c^2*x^2)*(7*pi^2*b*c^10*x^10 + 26*pi^2*b*c^8*x^8 + 34*pi^2*b*c^6*x^6 + 16*pi^2*b*c^4*x^4 - pi^2*b*c^2*x^2 - 2*pi^2*b)*log(c* x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2)*(441*pi^2*a*c^10*x^10 + 163 8*pi^2*a*c^8*x^8 + 2142*pi^2*a*c^6*x^6 + 1008*pi^2*a*c^4*x^4 - 63*pi^2*a*c ^2*x^2 - 126*pi^2*a - (49*pi^2*b*c^9*x^9 + 171*pi^2*b*c^7*x^7 + 189*pi^2*b *c^5*x^5 + 21*pi^2*b*c^3*x^3 - 126*pi^2*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^6*x^ 2 + c^4)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (133) = 266\).
Time = 92.06 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.69 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{8} \sqrt {c^{2} x^{2} + 1}}{9} + \frac {19 \pi ^{\frac {5}{2}} a c^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{63} + \frac {5 \pi ^{\frac {5}{2}} a x^{4} \sqrt {c^{2} x^{2} + 1}}{21} + \frac {\pi ^{\frac {5}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{63 c^{2}} - \frac {2 \pi ^{\frac {5}{2}} a \sqrt {c^{2} x^{2} + 1}}{63 c^{4}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{9}}{81} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {19 \pi ^{\frac {5}{2}} b c^{3} x^{7}}{441} + \frac {19 \pi ^{\frac {5}{2}} b c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63} - \frac {\pi ^{\frac {5}{2}} b c x^{5}}{21} + \frac {5 \pi ^{\frac {5}{2}} b x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{21} - \frac {\pi ^{\frac {5}{2}} b x^{3}}{189 c} + \frac {\pi ^{\frac {5}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63 c^{2}} + \frac {2 \pi ^{\frac {5}{2}} b x}{63 c^{3}} - \frac {2 \pi ^{\frac {5}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63 c^{4}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x)),x)
Output:
Piecewise((pi**(5/2)*a*c**4*x**8*sqrt(c**2*x**2 + 1)/9 + 19*pi**(5/2)*a*c* *2*x**6*sqrt(c**2*x**2 + 1)/63 + 5*pi**(5/2)*a*x**4*sqrt(c**2*x**2 + 1)/21 + pi**(5/2)*a*x**2*sqrt(c**2*x**2 + 1)/(63*c**2) - 2*pi**(5/2)*a*sqrt(c** 2*x**2 + 1)/(63*c**4) - pi**(5/2)*b*c**5*x**9/81 + pi**(5/2)*b*c**4*x**8*s qrt(c**2*x**2 + 1)*asinh(c*x)/9 - 19*pi**(5/2)*b*c**3*x**7/441 + 19*pi**(5 /2)*b*c**2*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/63 - pi**(5/2)*b*c*x**5/21 + 5*pi**(5/2)*b*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/21 - pi**(5/2)*b*x**3/ (189*c) + pi**(5/2)*b*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(63*c**2) + 2*pi **(5/2)*b*x/(63*c**3) - 2*pi**(5/2)*b*sqrt(c**2*x**2 + 1)*asinh(c*x)/(63*c **4), Ne(c, 0)), (pi**(5/2)*a*x**4/4, True))
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{63} \, {\left (\frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{4}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{4}}\right )} a - \frac {{\left (49 \, \pi ^{\frac {5}{2}} c^{8} x^{9} + 171 \, \pi ^{\frac {5}{2}} c^{6} x^{7} + 189 \, \pi ^{\frac {5}{2}} c^{4} x^{5} + 21 \, \pi ^{\frac {5}{2}} c^{2} x^{3} - 126 \, \pi ^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \] Input:
integrate(x^3*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxim a")
Output:
1/63*(7*(pi + pi*c^2*x^2)^(7/2)*x^2/(pi*c^2) - 2*(pi + pi*c^2*x^2)^(7/2)/( pi*c^4))*b*arcsinh(c*x) + 1/63*(7*(pi + pi*c^2*x^2)^(7/2)*x^2/(pi*c^2) - 2 *(pi + pi*c^2*x^2)^(7/2)/(pi*c^4))*a - 1/3969*(49*pi^(5/2)*c^8*x^9 + 171*p i^(5/2)*c^6*x^7 + 189*pi^(5/2)*c^4*x^5 + 21*pi^(5/2)*c^2*x^3 - 126*pi^(5/2 )*x)*b/c^3
Exception generated. \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),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
Timed out. \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \] Input:
int(x^3*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2),x)
Output:
int(x^3*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2), x)
\[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (7 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}+19 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+15 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a +63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+126 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+63 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{63 c^{4}} \] Input:
int(x^3*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(pi)*pi**2*(7*sqrt(c**2*x**2 + 1)*a*c**8*x**8 + 19*sqrt(c**2*x**2 + 1 )*a*c**6*x**6 + 15*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + sqrt(c**2*x**2 + 1)*a *c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a + 63*int(sqrt(c**2*x**2 + 1)*asinh(c* x)*x**7,x)*b*c**8 + 126*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6 + 63*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3,x)*b*c**4))/(63*c**4)