Integrand size = 26, antiderivative size = 173 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {8 b \pi ^{3/2} x}{315 c^5}+\frac {4 b \pi ^{3/2} x^3}{945 c^3}-\frac {b \pi ^{3/2} x^5}{525 c}-\frac {10}{441} b c \pi ^{3/2} x^7-\frac {1}{81} b c^3 \pi ^{3/2} x^9+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6 \pi }-\frac {2 \left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6 \pi ^3} \] Output:
-8/315*b*Pi^(3/2)*x/c^5+4/945*b*Pi^(3/2)*x^3/c^3-1/525*b*Pi^(3/2)*x^5/c-10 /441*b*c*Pi^(3/2)*x^7-1/81*b*c^3*Pi^(3/2)*x^9+1/5*(Pi*c^2*x^2+Pi)^(5/2)*(a +b*arcsinh(c*x))/c^6/Pi-2/7*(Pi*c^2*x^2+Pi)^(7/2)*(a+b*arcsinh(c*x))/c^6/P i^2+1/9*(Pi*c^2*x^2+Pi)^(9/2)*(a+b*arcsinh(c*x))/c^6/Pi^3
Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (315 a \left (1+c^2 x^2\right )^{5/2} \left (8-20 c^2 x^2+35 c^4 x^4\right )-b c x \left (2520-420 c^2 x^2+189 c^4 x^4+2250 c^6 x^6+1225 c^8 x^8\right )+315 b \left (1+c^2 x^2\right )^{5/2} \left (8-20 c^2 x^2+35 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{99225 c^6} \] Input:
Integrate[x^5*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
(Pi^(3/2)*(315*a*(1 + c^2*x^2)^(5/2)*(8 - 20*c^2*x^2 + 35*c^4*x^4) - b*c*x *(2520 - 420*c^2*x^2 + 189*c^4*x^4 + 2250*c^6*x^6 + 1225*c^8*x^8) + 315*b* (1 + c^2*x^2)^(5/2)*(8 - 20*c^2*x^2 + 35*c^4*x^4)*ArcSinh[c*x]))/(99225*c^ 6)
Time = 0.79 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 1467, 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.
| \(\displaystyle \int x^5 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int \frac {\pi \left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )}{315 c^6}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\pi ^{3/2} b \int \left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^6}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {\pi ^{3/2} b \int \left (35 c^8 x^8+50 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi c^6}-\frac {\pi ^{3/2} b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\) |
Input:
Int[x^5*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/315*(b*Pi^(3/2)*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (50*c^6*x^7)/7 + (35*c^8*x^9)/9))/c^5 + ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5* c^6*Pi) - (2*(Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^6*Pi^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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 = 1.12 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.25
| method | result | size |
| orering | \(\frac {\left (20825 c^{10} x^{10}+50900 c^{8} x^{8}+29457 c^{6} x^{6}-2730 c^{4} x^{4}+19320 c^{2} x^{2}+15120\right ) \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{99225 c^{6} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (1225 c^{8} x^{8}+2250 c^{6} x^{6}+189 c^{4} x^{4}-420 c^{2} x^{2}+2520\right ) \left (5 x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+3 x^{6} \sqrt {\pi \,c^{2} x^{2}+\pi }\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}+\frac {x^{5} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{99225 x^{4} c^{6} \left (c^{2} x^{2}+1\right )}\) | \(217\) |
| default | \(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{9 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{7 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{35 \pi \,c^{4}}\right )}{9 c^{2}}\right )+\frac {b \,\pi ^{\frac {3}{2}} \left (11025 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}+26775 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-1225 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+16695 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-2250 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-315 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-189 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+1260 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+420 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+2520 \,\operatorname {arcsinh}\left (x c \right )-2520 \sqrt {c^{2} x^{2}+1}\, x c \right )}{99225 c^{6} \sqrt {c^{2} x^{2}+1}}\) | \(255\) |
| parts | \(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{9 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{7 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{35 \pi \,c^{4}}\right )}{9 c^{2}}\right )+\frac {b \,\pi ^{\frac {3}{2}} \left (11025 \,\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}+26775 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}-1225 x^{9} c^{9} \sqrt {c^{2} x^{2}+1}+16695 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-2250 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-315 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-189 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+1260 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+420 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+2520 \,\operatorname {arcsinh}\left (x c \right )-2520 \sqrt {c^{2} x^{2}+1}\, x c \right )}{99225 c^{6} \sqrt {c^{2} x^{2}+1}}\) | \(255\) |
Input:
int(x^5*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/99225*(20825*c^10*x^10+50900*c^8*x^8+29457*c^6*x^6-2730*c^4*x^4+19320*c^ 2*x^2+15120)/c^6/(c^2*x^2+1)^2*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))-1/ 99225/x^4*(1225*c^8*x^8+2250*c^6*x^6+189*c^4*x^4-420*c^2*x^2+2520)/c^6/(c^ 2*x^2+1)*(5*x^4*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))+3*x^6*(Pi*c^2*x^2 +Pi)^(1/2)*(a+b*arcsinh(x*c))*Pi*c^2+x^5*(Pi*c^2*x^2+Pi)^(3/2)*b*c/(c^2*x^ 2+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.32 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {315 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (35 \, \pi b c^{10} x^{10} + 85 \, \pi b c^{8} x^{8} + 53 \, \pi b c^{6} x^{6} - \pi b c^{4} x^{4} + 4 \, \pi b c^{2} x^{2} + 8 \, \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (11025 \, \pi a c^{10} x^{10} + 26775 \, \pi a c^{8} x^{8} + 16695 \, \pi a c^{6} x^{6} - 315 \, \pi a c^{4} x^{4} + 1260 \, \pi a c^{2} x^{2} + 2520 \, \pi a - {\left (1225 \, \pi b c^{9} x^{9} + 2250 \, \pi b c^{7} x^{7} + 189 \, \pi b c^{5} x^{5} - 420 \, \pi b c^{3} x^{3} + 2520 \, \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{99225 \, {\left (c^{8} x^{2} + c^{6}\right )}} \] Input:
integrate(x^5*(pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="frica s")
Output:
1/99225*(315*sqrt(pi + pi*c^2*x^2)*(35*pi*b*c^10*x^10 + 85*pi*b*c^8*x^8 + 53*pi*b*c^6*x^6 - pi*b*c^4*x^4 + 4*pi*b*c^2*x^2 + 8*pi*b)*log(c*x + sqrt(c ^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2)*(11025*pi*a*c^10*x^10 + 26775*pi*a*c^ 8*x^8 + 16695*pi*a*c^6*x^6 - 315*pi*a*c^4*x^4 + 1260*pi*a*c^2*x^2 + 2520*p i*a - (1225*pi*b*c^9*x^9 + 2250*pi*b*c^7*x^7 + 189*pi*b*c^5*x^5 - 420*pi*b *c^3*x^3 + 2520*pi*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^8*x^2 + c^6)
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (167) = 334\).
Time = 29.40 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.20 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{8} \sqrt {c^{2} x^{2} + 1}}{9} + \frac {10 \pi ^{\frac {3}{2}} a x^{6} \sqrt {c^{2} x^{2} + 1}}{63} + \frac {\pi ^{\frac {3}{2}} a x^{4} \sqrt {c^{2} x^{2} + 1}}{105 c^{2}} - \frac {4 \pi ^{\frac {3}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{315 c^{4}} + \frac {8 \pi ^{\frac {3}{2}} a \sqrt {c^{2} x^{2} + 1}}{315 c^{6}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{9}}{81} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {10 \pi ^{\frac {3}{2}} b c x^{7}}{441} + \frac {10 \pi ^{\frac {3}{2}} b x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63} - \frac {\pi ^{\frac {3}{2}} b x^{5}}{525 c} + \frac {\pi ^{\frac {3}{2}} b x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{105 c^{2}} + \frac {4 \pi ^{\frac {3}{2}} b x^{3}}{945 c^{3}} - \frac {4 \pi ^{\frac {3}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{315 c^{4}} - \frac {8 \pi ^{\frac {3}{2}} b x}{315 c^{5}} + \frac {8 \pi ^{\frac {3}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{315 c^{6}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{6}}{6} & \text {otherwise} \end {cases} \] Input:
integrate(x**5*(pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x)),x)
Output:
Piecewise((pi**(3/2)*a*c**2*x**8*sqrt(c**2*x**2 + 1)/9 + 10*pi**(3/2)*a*x* *6*sqrt(c**2*x**2 + 1)/63 + pi**(3/2)*a*x**4*sqrt(c**2*x**2 + 1)/(105*c**2 ) - 4*pi**(3/2)*a*x**2*sqrt(c**2*x**2 + 1)/(315*c**4) + 8*pi**(3/2)*a*sqrt (c**2*x**2 + 1)/(315*c**6) - pi**(3/2)*b*c**3*x**9/81 + pi**(3/2)*b*c**2*x **8*sqrt(c**2*x**2 + 1)*asinh(c*x)/9 - 10*pi**(3/2)*b*c*x**7/441 + 10*pi** (3/2)*b*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/63 - pi**(3/2)*b*x**5/(525*c) + pi**(3/2)*b*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(105*c**2) + 4*pi**(3/2) *b*x**3/(945*c**3) - 4*pi**(3/2)*b*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(31 5*c**4) - 8*pi**(3/2)*b*x/(315*c**5) + 8*pi**(3/2)*b*sqrt(c**2*x**2 + 1)*a sinh(c*x)/(315*c**6), Ne(c, 0)), (pi**(3/2)*a*x**6/6, True))
Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.17 \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{315} \, {\left (\frac {35 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{4}}{\pi c^{2}} - \frac {20 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi c^{6}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{315} \, {\left (\frac {35 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{4}}{\pi c^{2}} - \frac {20 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}{\pi c^{6}}\right )} a - \frac {{\left (1225 \, \pi ^{\frac {3}{2}} c^{8} x^{9} + 2250 \, \pi ^{\frac {3}{2}} c^{6} x^{7} + 189 \, \pi ^{\frac {3}{2}} c^{4} x^{5} - 420 \, \pi ^{\frac {3}{2}} c^{2} x^{3} + 2520 \, \pi ^{\frac {3}{2}} x\right )} b}{99225 \, c^{5}} \] Input:
integrate(x^5*(pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="maxim a")
Output:
1/315*(35*(pi + pi*c^2*x^2)^(5/2)*x^4/(pi*c^2) - 20*(pi + pi*c^2*x^2)^(5/2 )*x^2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(5/2)/(pi*c^6))*b*arcsinh(c*x) + 1/31 5*(35*(pi + pi*c^2*x^2)^(5/2)*x^4/(pi*c^2) - 20*(pi + pi*c^2*x^2)^(5/2)*x^ 2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(5/2)/(pi*c^6))*a - 1/99225*(1225*pi^(3/2 )*c^8*x^9 + 2250*pi^(3/2)*c^6*x^7 + 189*pi^(3/2)*c^4*x^5 - 420*pi^(3/2)*c^ 2*x^3 + 2520*pi^(3/2)*x)*b/c^5
Exception generated. \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(pi*c^2*x^2+pi)^(3/2)*(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. \[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \] Input:
int(x^5*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2),x)
Output:
int(x^5*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2), x)
\[ \int x^5 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi \left (35 \sqrt {c^{2} x^{2}+1}\, a \,c^{8} x^{8}+50 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a +315 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+315 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{315 c^{6}} \] Input:
int(x^5*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(pi)*pi*(35*sqrt(c**2*x**2 + 1)*a*c**8*x**8 + 50*sqrt(c**2*x**2 + 1)* a*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 4*sqrt(c**2*x**2 + 1)*a* c**2*x**2 + 8*sqrt(c**2*x**2 + 1)*a + 315*int(sqrt(c**2*x**2 + 1)*asinh(c* x)*x**7,x)*b*c**8 + 315*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6) )/(315*c**6)