Integrand size = 26, antiderivative size = 139 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^5 \pi ^{5/2}}+\frac {2 b \log \left (1+c^2 x^2\right )}{3 c^5 \pi ^{5/2}} \]
1/6*b/c^5/Pi^(5/2)/(c^2*x^2+1)-1/3*x^3*(a+b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x ^2+Pi)^(3/2)+1/2*(a+b*arcsinh(c*x))^2/b/c^5/Pi^(5/2)+2/3*b*ln(c^2*x^2+1)/c ^5/Pi^(5/2)-x*(a+b*arcsinh(c*x))/c^4/Pi^2/(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b+b c^2 x^2-6 a c x \sqrt {1+c^2 x^2}-8 a c^3 x^3 \sqrt {1+c^2 x^2}+2 \left (3 a \left (1+c^2 x^2\right )^2-b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )\right ) \text {arcsinh}(c x)+3 b \left (1+c^2 x^2\right )^2 \text {arcsinh}(c x)^2+4 b \left (1+c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )^2} \]
(b + b*c^2*x^2 - 6*a*c*x*Sqrt[1 + c^2*x^2] - 8*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2))*ArcSi nh[c*x] + 3*b*(1 + c^2*x^2)^2*ArcSinh[c*x]^2 + 4*b*(1 + c^2*x^2)^2*Log[1 + c^2*x^2])/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)^2)
Time = 0.75 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6225, 243, 49, 2009, 6225, 240, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{\pi c^2}+\frac {b \int \frac {x^3}{\left (c^2 x^2+1\right )^2}dx}{3 \pi ^{5/2} c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{\pi c^2}+\frac {b \int \frac {x^2}{\left (c^2 x^2+1\right )^2}dx^2}{6 \pi ^{5/2} c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{\pi c^2}+\frac {b \int \left (\frac {1}{c^2 \left (c^2 x^2+1\right )}-\frac {1}{c^2 \left (c^2 x^2+1\right )^2}\right )dx^2}{6 \pi ^{5/2} c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 \pi x^2+\pi \right )^{3/2}}dx}{\pi c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 \pi ^{5/2} c}\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}+\frac {b \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2} c}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}}{\pi c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 \pi ^{5/2} c}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{\pi c^2}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}}{\pi c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 \pi ^{5/2} c}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {\frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{3/2} b c^3}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3}}{\pi c^2}+\frac {b \left (\frac {1}{c^4 \left (c^2 x^2+1\right )}+\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )}{6 \pi ^{5/2} c}\) |
-1/3*(x^3*(a + b*ArcSinh[c*x]))/(c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (b*(1/( c^4*(1 + c^2*x^2)) + Log[1 + c^2*x^2]/c^4))/(6*c*Pi^(5/2)) + (-((x*(a + b* ArcSinh[c*x]))/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (a + b*ArcSinh[c*x])^2/(2 *b*c^3*Pi^(3/2)) + (b*Log[1 + c^2*x^2])/(2*c^3*Pi^(3/2)))/(c^2*Pi)
3.2.2.3.1 Defintions of rubi rules used
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]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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]
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]
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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(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]) /; Fre eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(896\) vs. \(2(121)=242\).
Time = 0.17 (sec) , antiderivative size = 897, normalized size of antiderivative = 6.45
| method | result | size |
| default | \(-\frac {a \,x^{3}}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {a x}{\pi ^{2} c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi ^{2} c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{5} \pi ^{\frac {5}{2}}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 c^{5} \pi ^{\frac {5}{2}}}+\frac {32 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{3} x^{8}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )}+\frac {116 b c \,\operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {76 b \,\operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {4 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c}+\frac {472 b \,\operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {181 b \,\operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {16 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {3 b \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c^{3}}+\frac {284 b \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {16 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {32 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}+\frac {64 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {8 b}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {4 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{5} \pi ^{\frac {5}{2}}}\) | \(897\) |
| parts | \(-\frac {a \,x^{3}}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {a x}{\pi ^{2} c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi ^{2} c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{5} \pi ^{\frac {5}{2}}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 c^{5} \pi ^{\frac {5}{2}}}+\frac {32 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{3} x^{8}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )}+\frac {116 b c \,\operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {76 b \,\operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {4 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c}+\frac {472 b \,\operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {181 b \,\operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {16 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {3 b \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c^{3}}+\frac {284 b \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {16 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {32 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}+\frac {64 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {8 b}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {4 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{5} \pi ^{\frac {5}{2}}}\) | \(897\) |
-1/3*a*x^3/Pi/c^2/(Pi*c^2*x^2+Pi)^(3/2)-a/Pi^2/c^4*x/(Pi*c^2*x^2+Pi)^(1/2) +a/Pi^2/c^4*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/ 2)+1/2*b/c^5/Pi^(5/2)*arcsinh(c*x)^2-8/3*b/c^5/Pi^(5/2)*arcsinh(c*x)+32*b/ Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c^3*arcsinh(c*x)*x^8-32* b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^(3/2)*c^2*arcsinh(c*x)*x ^7+8/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c^3*x^8-8/3*b/P i^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)*c*x^6+116*b/Pi^(5/2)/(24*c^ 4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c*arcsinh(c*x)*x^6-76*b/Pi^(5/2)/(24*c^ 4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)*x^5+32/3*b/Pi^(5/2)/(2 4*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c*x^6-4*b/Pi^(5/2)/(24*c^4*x^4+39*c ^2*x^2+16)/(c^2*x^2+1)/c*x^4+472/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/( c^2*x^2+1)^2/c*arcsinh(c*x)*x^4-181/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16 )/(c^2*x^2+1)^(3/2)/c^2*arcsinh(c*x)*x^3+16*b/Pi^(5/2)/(24*c^4*x^4+39*c^2* x^2+16)/(c^2*x^2+1)^2/c*x^4-3/2*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2 *x^2+1)/c^3*x^2+284/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/ c^3*arcsinh(c*x)*x^2-16*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^ (3/2)/c^4*arcsinh(c*x)*x+32/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x ^2+1)^2/c^3*x^2+64/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c ^5*arcsinh(c*x)+8/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c^ 5+4/3*b/c^5/Pi^(5/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(b*x^4*arcsinh(c*x) + a*x^4)/(pi^3*c^6*x^6 + 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 + pi^3), x)
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a x^{4}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
(Integral(a*x**4/(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**4*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)
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*(x*(3*x^2/(pi*(pi + pi*c^2*x^2)^(3/2)*c^2) + 2/(pi*(pi + pi*c^2*x^2)^ (3/2)*c^4)) + x/(pi^2*sqrt(pi + pi*c^2*x^2)*c^4) - 3*arcsinh(c*x)/(pi^(5/2 )*c^5))*a + b*integrate(x^4*log(c*x + sqrt(c^2*x^2 + 1))/(pi + pi*c^2*x^2) ^(5/2), x)
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]