\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{5/2}} \, dx\) [114]

Optimal result

Integrand size = 26, antiderivative size = 80 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^3 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{6 c^3 \pi ^{5/2}} \] Output:

-1/6*b/c^3/Pi^(5/2)/(c^2*x^2+1)+1/3*x^3*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+ 
Pi)^(3/2)-1/6*b*ln(c^2*x^2+1)/c^3/Pi^(5/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {-2 a c^3 x^3+b \sqrt {1+c^2 x^2}-2 b c^3 x^3 \text {arcsinh}(c x)+b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{6 c^3 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \] Input:

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

Output:

-1/6*(-2*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] - 2*b*c^3*x^3*ArcSinh[c*x] + b*(1 
 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(c^3*Pi^(5/2)*(1 + c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6215, 243, 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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x^3*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (b*c*(1/(c^4*( 
1 + c^2*x^2)) + Log[1 + c^2*x^2]/c^4))/(6*Pi^(5/2))
 

Defintions of rubi rules used

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 6215

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 + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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, p}, x] && EqQ 
[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(68)=136\).

Time = 1.11 (sec) , antiderivative size = 729, normalized size of antiderivative = 9.11

method	result	size
default	\(a \left (-\frac {x}{2 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}}{2 c^{2}}\right )+\frac {2 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c^{3}}-\frac {b \,c^{5} \operatorname {arcsinh}\left (x c \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{4} \operatorname {arcsinh}\left (x c \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{5} x^{8}}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{3} x^{6}}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )}-\frac {3 b \,c^{3} \operatorname {arcsinh}\left (x c \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{2} \operatorname {arcsinh}\left (x c \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 b \,c^{3} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {10 b c \,\operatorname {arcsinh}\left (x c \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,\operatorname {arcsinh}\left (x c \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c \,x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {5 b \,\operatorname {arcsinh}\left (x c \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {2 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {b}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {b \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}} c^{3}}\)	\(729\)
parts	\(a \left (-\frac {x}{2 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}}{2 c^{2}}\right )+\frac {2 b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} c^{3}}-\frac {b \,c^{5} \operatorname {arcsinh}\left (x c \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{4} \operatorname {arcsinh}\left (x c \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{5} x^{8}}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{3} x^{6}}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )}-\frac {3 b \,c^{3} \operatorname {arcsinh}\left (x c \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,c^{2} \operatorname {arcsinh}\left (x c \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 b \,c^{3} x^{6}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {10 b c \,\operatorname {arcsinh}\left (x c \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \,\operatorname {arcsinh}\left (x c \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c \,x^{4}}{\pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {5 b \,\operatorname {arcsinh}\left (x c \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {2 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {b \,\operatorname {arcsinh}\left (x c \right )}{3 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {b}{6 \pi ^{\frac {5}{2}} \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {b \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}} c^{3}}\)	\(729\)

Input:

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

Output:

a*(-1/2*x/Pi/c^2/(Pi*c^2*x^2+Pi)^(3/2)+1/2/c^2*(1/3/Pi*x/(Pi*c^2*x^2+Pi)^( 
3/2)+2/3/Pi^2*x/(Pi*c^2*x^2+Pi)^(1/2)))+2/3*b/Pi^(5/2)/c^3*arcsinh(x*c)-b/ 
Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2*c^5*arcsinh(x*c)*x^8+b/Pi^( 
5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(3/2)*c^4*arcsinh(x*c)*x^7-1/6*b/ 
Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2*c^5*x^8+1/6*b/Pi^(5/2)/(3*c 
^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)*c^3*x^6-3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+ 
1)/(c^2*x^2+1)^2*c^3*arcsinh(x*c)*x^6+b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/( 
c^2*x^2+1)^(3/2)*c^2*arcsinh(x*c)*x^5-2/3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+ 
1)/(c^2*x^2+1)^2*c^3*x^6-10/3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+ 
1)^2*c*arcsinh(x*c)*x^4+1/3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1) 
^(3/2)*arcsinh(x*c)*x^3-b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2*c 
*x^4-5/3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2/c*arcsinh(x*c)*x 
^2-2/3*b/Pi^(5/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2/c*x^2-1/3*b/Pi^(5/ 
2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2/c^3*arcsinh(x*c)-1/6*b/Pi^(5/2)/( 
3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^2/c^3-1/3*b/Pi^(5/2)/c^3*ln(1+(x*c+(c^2 
*x^2+1)^(1/2))^2)
 

Fricas [F]

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

integrate(x^2*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="frica 
s")
 

Output:

integral(sqrt(pi + pi*c^2*x^2)*(b*x^2*arcsinh(c*x) + a*x^2)/(pi^3*c^6*x^6 
+ 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 + pi^3), x)
 

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a x^{2}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \] Input:

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

Output:

(Integral(a*x**2/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x* 
*2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x**2*asinh(c*x)/(c**4*x**4 
*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 
1)), x))/pi**(5/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (68) = 136\).

Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.71 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {1}{6} \, b c {\left (\frac {1}{\pi ^{\frac {5}{2}} c^{6} x^{2} + \pi ^{\frac {5}{2}} c^{4}} + \frac {\log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {5}{2}} c^{4}}\right )} - \frac {1}{3} \, b {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}} - \frac {x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}} c^{2}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}} - \frac {x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}} c^{2}}\right )} \] Input:

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

Output:

-1/6*b*c*(1/(pi^(5/2)*c^6*x^2 + pi^(5/2)*c^4) + log(c^2*x^2 + 1)/(pi^(5/2) 
*c^4)) - 1/3*b*(x/(pi*(pi + pi*c^2*x^2)^(3/2)*c^2) - x/(pi^2*sqrt(pi + pi* 
c^2*x^2)*c^2))*arcsinh(c*x) - 1/3*a*(x/(pi*(pi + pi*c^2*x^2)^(3/2)*c^2) - 
x/(pi^2*sqrt(pi + pi*c^2*x^2)*c^2))
 

Giac [F]

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

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

Output:

integrate((b*arcsinh(c*x) + a)*x^2/(pi + pi*c^2*x^2)^(5/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{7} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{5} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3}+a \,c^{4} x^{4}+2 a \,c^{2} x^{2}+a}{3 \sqrt {\pi }\, c^{3} \pi ^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 3*int((asinh(c*x)*x**2)/(sqrt(c**2*x**2 
 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x 
)*b*c**7*x**4 + 6*int((asinh(c*x)*x**2)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2 
*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**5*x**2 + 3*i 
nt((asinh(c*x)*x**2)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1 
)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**3 + a*c**4*x**4 + 2*a*c**2*x**2 
 + a)/(3*sqrt(pi)*c**3*pi**2*(c**4*x**4 + 2*c**2*x**2 + 1))