Integrand size = 26, antiderivative size = 166 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c \pi ^{5/2}}{6 x^2}-\frac {1}{4} b c^5 \pi ^{5/2} x^2+\frac {5}{2} c^4 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {5 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {5 c^3 \pi ^{5/2} (a+b \text {arcsinh}(c x))^2}{4 b}+\frac {7}{3} b c^3 \pi ^{5/2} \log (x) \]
-1/6*b*c*Pi^(5/2)/x^2-1/4*b*c^5*Pi^(5/2)*x^2-5/3*c^2*Pi*(Pi*c^2*x^2+Pi)^(3 /2)*(a+b*arcsinh(c*x))/x-1/3*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/x^3+ 5/4*c^3*Pi^(5/2)*(a+b*arcsinh(c*x))^2/b+7/3*b*c^3*Pi^(5/2)*ln(x)+5/2*c^4*P i^2*x*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.48 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.08 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\pi ^{5/2} \left (-4 b c x-8 a \sqrt {1+c^2 x^2}-56 a c^2 x^2 \sqrt {1+c^2 x^2}+12 a c^4 x^4 \sqrt {1+c^2 x^2}+30 b c^3 x^3 \text {arcsinh}(c x)^2-3 b c^3 x^3 \cosh (2 \text {arcsinh}(c x))+56 b c^3 x^3 \log (c x)+\text {arcsinh}(c x) \left (60 a c^3 x^3-8 b \sqrt {1+c^2 x^2} \left (1+7 c^2 x^2\right )+6 b c^3 x^3 \sinh (2 \text {arcsinh}(c x))\right )\right )}{24 x^3} \]
(Pi^(5/2)*(-4*b*c*x - 8*a*Sqrt[1 + c^2*x^2] - 56*a*c^2*x^2*Sqrt[1 + c^2*x^ 2] + 12*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 30*b*c^3*x^3*ArcSinh[c*x]^2 - 3*b*c^ 3*x^3*Cosh[2*ArcSinh[c*x]] + 56*b*c^3*x^3*Log[c*x] + ArcSinh[c*x]*(60*a*c^ 3*x^3 - 8*b*Sqrt[1 + c^2*x^2]*(1 + 7*c^2*x^2) + 6*b*c^3*x^3*Sinh[2*ArcSinh [c*x]])))/(24*x^3)
Time = 0.86 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6222, 243, 49, 2009, 6222, 244, 2009, 6200, 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.
| \(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {5}{3} \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{3} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^3}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{3} \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{6} \pi ^{5/2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^4}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {5}{3} \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{6} \pi ^{5/2} b c \int \left (c^4+\frac {2 c^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{3} \pi c^2 \int \frac {\left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\pi ^{3/2} b c \int \frac {c^2 x^2+1}{x}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\pi ^{3/2} b c \int \left (x c^2+\frac {1}{x}\right )dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} \sqrt {\pi } b c \int xdx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \left (\frac {1}{2} \sqrt {\pi } \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{4} \sqrt {\pi } b c x^2\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {5}{3} \pi c^2 \left (3 \pi c^2 \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} \sqrt {\pi } b c x^2\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\pi ^{3/2} b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{6} \pi ^{5/2} b c \left (c^4 x^2+2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )\) |
-1/3*((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3 + (5*c^2*Pi*(-(((P i + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x) + 3*c^2*Pi*(-1/4*(b*c*Sqrt[ Pi]*x^2) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(4*b*c)) + b*c*Pi^(3/2)*((c^2*x^2)/2 + Log[x])))/3 + (b*c*Pi^(5/2)*(-x^(-2) + c^4*x^2 + 2*c^2*Log[x^2]))/6
3.1.78.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_)^(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[((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]
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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
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 + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], 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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(691\) vs. \(2(138)=276\).
Time = 0.22 (sec) , antiderivative size = 692, normalized size of antiderivative = 4.17
| method | result | size |
| default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{3 \pi \,x^{3}}-\frac {4 a \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{3 \pi x}+\frac {4 a \,c^{4} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {5 a \,c^{4} \pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}-\frac {b \,c^{3} \pi ^{\frac {5}{2}}}{8}-\frac {14 b \,c^{3} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )}{3}+\frac {5 b \,c^{3} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )^{2}}{4}+\frac {147 b \,\pi ^{\frac {5}{2}} x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {49 b \,\pi ^{\frac {5}{2}} x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}+\frac {35 b \,\pi ^{\frac {5}{2}} x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {7 b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}+\frac {7 b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}-\frac {b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) c}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x \,c^{4}}{2}+\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{2}}{4}+\frac {49 b \,\pi ^{\frac {5}{2}} x^{4} c^{7}}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}-\frac {147 b \,\pi ^{\frac {5}{2}} x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {56 b \,\pi ^{\frac {5}{2}} x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {22 b \,\pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x}\) | \(692\) |
| parts | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{3 \pi \,x^{3}}-\frac {4 a \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{3 \pi x}+\frac {4 a \,c^{4} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {5 a \,c^{4} \pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}-\frac {b \,c^{3} \pi ^{\frac {5}{2}}}{8}-\frac {14 b \,c^{3} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )}{3}+\frac {5 b \,c^{3} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )^{2}}{4}+\frac {147 b \,\pi ^{\frac {5}{2}} x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {49 b \,\pi ^{\frac {5}{2}} x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}+\frac {35 b \,\pi ^{\frac {5}{2}} x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {7 b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}+\frac {7 b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}-\frac {b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right ) c}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x \,c^{4}}{2}+\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{2}}{4}+\frac {49 b \,\pi ^{\frac {5}{2}} x^{4} c^{7}}{6 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right )}-\frac {147 b \,\pi ^{\frac {5}{2}} x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {56 b \,\pi ^{\frac {5}{2}} x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{63 c^{4} x^{4}+15 c^{2} x^{2}+1}-\frac {22 b \,\pi ^{\frac {5}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (63 c^{4} x^{4}+15 c^{2} x^{2}+1\right ) x}\) | \(692\) |
-1/3*a/Pi/x^3*(Pi*c^2*x^2+Pi)^(7/2)-4/3*a*c^2/Pi/x*(Pi*c^2*x^2+Pi)^(7/2)+4 /3*a*c^4*x*(Pi*c^2*x^2+Pi)^(5/2)+5/3*a*c^4*Pi*x*(Pi*c^2*x^2+Pi)^(3/2)+5/2* a*c^4*Pi^2*x*(Pi*c^2*x^2+Pi)^(1/2)+5/2*a*c^4*Pi^3*ln(Pi*c^2*x/(Pi*c^2)^(1/ 2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)-1/8*b*c^3*Pi^(5/2)-14/3*b*c^3*Pi^ (5/2)*arcsinh(c*x)+5/4*b*c^3*Pi^(5/2)*arcsinh(c*x)^2+147*b*Pi^(5/2)/(63*c^ 4*x^4+15*c^2*x^2+1)*x^4*arcsinh(c*x)*c^7-49/6*b*Pi^(5/2)/(63*c^4*x^4+15*c^ 2*x^2+1)*x^2*(c^2*x^2+1)*c^5+35*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*x^2*a rcsinh(c*x)*c^5-7/3*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*(c^2*x^2+1)*c^3+7 /3*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*arcsinh(c*x)*c^3-1/6*b*Pi^(5/2)/(6 3*c^4*x^4+15*c^2*x^2+1)/x^2*(c^2*x^2+1)*c-1/3*b*Pi^(5/2)/(63*c^4*x^4+15*c^ 2*x^2+1)/x^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)+1/2*b*(c^2*x^2+1)^(1/2)*arcsin h(c*x)*Pi^(5/2)*x*c^4+7/3*b*c^3*Pi^(5/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)-1 /4*b*c^5*Pi^(5/2)*x^2+49/6*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*x^4*c^7-14 7*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*x^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)* c^6-56*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)*x*(c^2*x^2+1)^(1/2)*arcsinh(c* x)*c^4-22/3*b*Pi^(5/2)/(63*c^4*x^4+15*c^2*x^2+1)/x*(c^2*x^2+1)^(1/2)*arcsi nh(c*x)*c^2
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(pi^2*a*c^4*x^4 + 2*pi^2*a*c^2*x^2 + pi^2*a + (pi^2*b*c^4*x^4 + 2*pi^2*b*c^2*x^2 + pi^2*b)*arcsinh(c*x))/x^4, x)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\pi ^{\frac {5}{2}} \left (\int a c^{4} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {2 a c^{2} \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int b c^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
pi**(5/2)*(Integral(a*c**4*sqrt(c**2*x**2 + 1), x) + Integral(a*sqrt(c**2* x**2 + 1)/x**4, x) + Integral(2*a*c**2*sqrt(c**2*x**2 + 1)/x**2, x) + Inte gral(b*c**4*sqrt(c**2*x**2 + 1)*asinh(c*x), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**4, x) + Integral(2*b*c**2*sqrt(c**2*x**2 + 1)*asinh(c* x)/x**2, x))
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^4} \,d x \]