\(\int x^2 (\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [70]

Optimal result

Integrand size = 26, antiderivative size = 162 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3} \] Output:

-1/32*b*Pi^(3/2)*x^2/c-7/96*b*c*Pi^(3/2)*x^4-1/36*b*c^3*Pi^(3/2)*x^6+1/16* 
Pi*x*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/c^2+1/8*Pi*x^3*(Pi*c^2*x^2+P 
i)^(1/2)*(a+b*arcsinh(c*x))+1/6*x^3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x 
))-1/32*Pi^(3/2)*(a+b*arcsinh(c*x))^2/b/c^3
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (144 a c x \sqrt {1+c^2 x^2}+672 a c^3 x^3 \sqrt {1+c^2 x^2}+384 a c^5 x^5 \sqrt {1+c^2 x^2}-72 b \text {arcsinh}(c x)^2+18 b \cosh (2 \text {arcsinh}(c x))-9 b \cosh (4 \text {arcsinh}(c x))-2 b \cosh (6 \text {arcsinh}(c x))-12 \text {arcsinh}(c x) (12 a+3 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))-b \sinh (6 \text {arcsinh}(c x)))\right )}{2304 c^3} \] Input:

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

Output:

(Pi^(3/2)*(144*a*c*x*Sqrt[1 + c^2*x^2] + 672*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 
 384*a*c^5*x^5*Sqrt[1 + c^2*x^2] - 72*b*ArcSinh[c*x]^2 + 18*b*Cosh[2*ArcSi 
nh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 12*ArcSin 
h[c*x]*(12*a + 3*b*Sinh[2*ArcSinh[c*x]] - 3*b*Sinh[4*ArcSinh[c*x]] - b*Sin 
h[6*ArcSinh[c*x]])))/(2304*c^3)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6223, 244, 2009, 6221, 15, 6227, 15, 6198}

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.

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

\(\Big \downarrow \) 6223

\(\displaystyle \frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{6} \pi ^{3/2} b c \int x^3 \left (c^2 x^2+1\right )dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{6} \pi ^{3/2} b c \int \left (c^2 x^5+x^3\right )dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

\(\Big \downarrow \) 6221

\(\displaystyle \frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{4} \sqrt {\pi } b c \int x^3dx+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} \pi \left (\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\frac {1}{4} \sqrt {\pi } \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )-\frac {1}{16} \sqrt {\pi } b c x^4\right )-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\)

Input:

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

Output:

-1/6*(b*c*Pi^(3/2)*(x^4/4 + (c^2*x^6)/6)) + (x^3*(Pi + c^2*Pi*x^2)^(3/2)*( 
a + b*ArcSinh[c*x]))/6 + (Pi*(-1/16*(b*c*Sqrt[Pi]*x^4) + (x^3*Sqrt[Pi + c^ 
2*Pi*x^2]*(a + b*ArcSinh[c*x]))/4 + (Sqrt[Pi]*(-1/4*(b*x^2)/c + (x*Sqrt[1 
+ c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3 
)))/4))/2
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6223

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m 
 + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + 
c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, 
 d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (x c \right )^{2}-4\right )}{288 c^{3}}\)	\(214\)
parts	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (x c \right )^{2}-4\right )}{288 c^{3}}\)	\(214\)

Input:

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

Output:

1/6*a*x*(Pi*c^2*x^2+Pi)^(5/2)/Pi/c^2-1/24*a/c^2*x*(Pi*c^2*x^2+Pi)^(3/2)-1/ 
16*a/c^2*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)-1/16*a/c^2*Pi^2*ln(Pi*c^2*x/(Pi*c^2)^( 
1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)-1/288*b*Pi^(3/2)*(-48*(c^2*x^2+ 
1)^(1/2)*arcsinh(x*c)*x^5*c^5+8*c^6*x^6-84*arcsinh(x*c)*(c^2*x^2+1)^(1/2)* 
x^3*c^3+21*c^4*x^4-18*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+9*c^2*x^2+9*arcsi 
nh(x*c)^2-4)/c^3
 

Fricas [F]

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

integrate(x^2*(pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="frica 
s")
 

Output:

integral(sqrt(pi + pi*c^2*x^2)*(pi*a*c^2*x^4 + pi*a*x^2 + (pi*b*c^2*x^4 + 
pi*b*x^2)*arcsinh(c*x)), x)
 

Sympy [A] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.62 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{6} + \frac {7 \pi ^{\frac {3}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{24} + \frac {\pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{6}}{36} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {7 \pi ^{\frac {3}{2}} b c x^{4}}{96} + \frac {7 \pi ^{\frac {3}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{24} - \frac {\pi ^{\frac {3}{2}} b x^{2}}{32 c} + \frac {\pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((pi**(3/2)*a*c**2*x**5*sqrt(c**2*x**2 + 1)/6 + 7*pi**(3/2)*a*x** 
3*sqrt(c**2*x**2 + 1)/24 + pi**(3/2)*a*x*sqrt(c**2*x**2 + 1)/(16*c**2) - p 
i**(3/2)*a*asinh(c*x)/(16*c**3) - pi**(3/2)*b*c**3*x**6/36 + pi**(3/2)*b*c 
**2*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/6 - 7*pi**(3/2)*b*c*x**4/96 + 7*pi 
**(3/2)*b*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/24 - pi**(3/2)*b*x**2/(32*c) 
 + pi**(3/2)*b*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(16*c**2) - pi**(3/2)*b*as 
inh(c*x)**2/(32*c**3), Ne(c, 0)), (pi**(3/2)*a*x**3/3, True))
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

integrate((pi + pi*c^2*x^2)^(3/2)*(b*arcsinh(c*x) + a)*x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi \left (8 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} x^{5}+14 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+3 \sqrt {c^{2} x^{2}+1}\, a c x +48 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{5}+48 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \right )}{48 c^{3}} \] Input:

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

Output:

(sqrt(pi)*pi*(8*sqrt(c**2*x**2 + 1)*a*c**5*x**5 + 14*sqrt(c**2*x**2 + 1)*a 
*c**3*x**3 + 3*sqrt(c**2*x**2 + 1)*a*c*x + 48*int(sqrt(c**2*x**2 + 1)*asin 
h(c*x)*x**4,x)*b*c**5 + 48*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b*c* 
*3 - 3*log(sqrt(c**2*x**2 + 1) + c*x)*a))/(48*c**3)