Integrand size = 26, antiderivative size = 150 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b c}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b c \log (x)}{\pi ^{5/2}}+\frac {5 b c \log \left (1+c^2 x^2\right )}{6 \pi ^{5/2}} \]
-1/6*b*c/Pi^(5/2)/(c^2*x^2+1)+(-a-b*arcsinh(c*x))/Pi/x/(Pi*c^2*x^2+Pi)^(3/ 2)-4/3*c^2*x*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+Pi)^(3/2)+b*c*ln(x)/Pi^(5/2 )+5/6*b*c*ln(c^2*x^2+1)/Pi^(5/2)-8/3*c^2*x*(a+b*arcsinh(c*x))/Pi^2/(Pi*c^2 *x^2+Pi)^(1/2)
Time = 0.41 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-b c x \sqrt {1+c^2 x^2}-2 a \left (3+12 c^2 x^2+8 c^4 x^4\right )-2 b \left (3+12 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+b c x \left (1+c^2 x^2\right )^{3/2} \left (16+6 \log (x)+5 \log \left (1+c^2 x^2\right )\right )}{6 \pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}} \]
(-(b*c*x*Sqrt[1 + c^2*x^2]) - 2*a*(3 + 12*c^2*x^2 + 8*c^4*x^4) - 2*b*(3 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] + b*c*x*(1 + c^2*x^2)^(3/2)*(16 + 6*L og[x] + 5*Log[1 + c^2*x^2]))/(6*Pi^(5/2)*x*(1 + c^2*x^2)^(3/2))
Time = 0.46 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\sqrt {\pi } b c \int -\frac {8 c^4 x^4+12 c^2 x^2+3}{3 \pi ^3 x \left (c^2 x^2+1\right )^2}dx-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {8 c^4 x^4+12 c^2 x^2+3}{x \left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c \int \frac {8 c^4 x^4+12 c^2 x^2+3}{x^2 \left (c^2 x^2+1\right )^2}dx^2}{6 \pi ^{5/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c \int \left (\frac {5 c^2}{c^2 x^2+1}+\frac {c^2}{\left (c^2 x^2+1\right )^2}+\frac {3}{x^2}\right )dx^2}{6 \pi ^{5/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )}{6 \pi ^{5/2}}\) |
-((a + b*ArcSinh[c*x])/(Pi*x*(Pi + c^2*Pi*x^2)^(3/2))) - (4*c^2*x*(a + b*A rcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x ]))/(3*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*(-(1 + c^2*x^2)^(-1) + 3*Log[x^2 ] + 5*Log[1 + c^2*x^2]))/(6*Pi^(5/2))
3.2.8.3.1 Defintions of rubi rules used
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_))^(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]
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]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(777\) vs. \(2(132)=264\).
Time = 0.18 (sec) , antiderivative size = 778, normalized size of antiderivative = 5.19
| method | result | size |
| default | \(a \left (-\frac {1}{\pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-4 c^{2} \left (\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 }}\right )\right )-\frac {16 b c \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}}}+\frac {32 b \,x^{10} c^{11}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {128 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{6} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {64 b \,x^{5} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {64 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{4} c^{5}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {200 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {56 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {128 b \,x^{4} c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {12 b \,x^{2} c^{3}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {208 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {44 b x \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b \,x^{2} c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c}{2 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {24 b \,\operatorname {arcsinh}\left (c x \right ) c}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 b \,\operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x}+\frac {5 b c \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{\pi ^{\frac {5}{2}}}\) | \(778\) |
| parts | \(a \left (-\frac {1}{\pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-4 c^{2} \left (\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 }}\right )\right )-\frac {16 b c \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}}}+\frac {32 b \,x^{10} c^{11}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {128 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{6} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {64 b \,x^{5} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {64 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{4} c^{5}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {200 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {56 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {128 b \,x^{4} c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {12 b \,x^{2} c^{3}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {208 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {44 b x \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b \,x^{2} c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c}{2 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {24 b \,\operatorname {arcsinh}\left (c x \right ) c}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 b \,\operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x}+\frac {5 b c \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{\pi ^{\frac {5}{2}}}\) | \(778\) |
a*(-1/Pi/x/(Pi*c^2*x^2+Pi)^(3/2)-4*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)))-16/3*b*c/Pi^(5/2)*arcsinh(c*x)+32/3*b/Pi^( 5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^10*c^11-32/3*b/Pi^(5/2)/(8*c^2*x^2+9)/( c^2*x^2+1)*x^8*c^9+128/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^8*c^9-32 *b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*x^6*c^7+64/3*b/Pi^(5/2)/(8*c^2*x^2+9 )/(c^2*x^2+1)^2*x^6*arcsinh(c*x)*c^7-64/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^ 2+1)^(3/2)*x^5*arcsinh(c*x)*c^6+64*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2* x^6*c^7-32*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*x^4*c^5+200/3*b/Pi^(5/2)/( 8*c^2*x^2+9)/(c^2*x^2+1)^2*x^4*arcsinh(c*x)*c^5-56*b/Pi^(5/2)/(8*c^2*x^2+9 )/(c^2*x^2+1)^(3/2)*x^3*arcsinh(c*x)*c^4+128/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c ^2*x^2+1)^2*x^4*c^5-12*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*x^2*c^3+208/3* b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^2*arcsinh(c*x)*c^3-44*b/Pi^(5/2)/ (8*c^2*x^2+9)/(c^2*x^2+1)^(3/2)*x*arcsinh(c*x)*c^2+32/3*b/Pi^(5/2)/(8*c^2* x^2+9)/(c^2*x^2+1)^2*x^2*c^3-3/2*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*c+24 *b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*arcsinh(c*x)*c-9*b/Pi^(5/2)/(8*c^2 *x^2+9)/(c^2*x^2+1)^(3/2)/x*arcsinh(c*x)+5/3*b*c/Pi^(5/2)*ln(1+(c*x+(c^2*x ^2+1)^(1/2))^2)+b*c/Pi^(5/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^3*c^6*x^8 + 3*pi^3 *c^4*x^6 + 3*pi^3*c^2*x^4 + pi^3*x^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
(Integral(a/(c**4*x**6*sqrt(c**2*x**2 + 1) + 2*c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(c**4*x**6*sqrt (c**2*x**2 + 1) + 2*c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x**2 + 1)), x))/pi**(5/2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
-1/3*a*(4*c^2*x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 8*c^2*x/(pi^2*sqrt(pi + pi* c^2*x^2)) + 3/(pi*(pi + pi*c^2*x^2)^(3/2)*x)) + b*integrate(log(c*x + sqrt (c^2*x^2 + 1))/((pi + pi*c^2*x^2)^(5/2)*x^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]