Integrand size = 25, antiderivative size = 275 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2} \]
-x*(c^2*x^2+1)^3/b/c/(a+b*arcsinh(c*x))+5/64*Chi((a+b*arcsinh(c*x))/b)*cos h(a/b)/b^2/c^2+27/64*Chi(3*(a+b*arcsinh(c*x))/b)*cosh(3*a/b)/b^2/c^2+25/64 *Chi(5*(a+b*arcsinh(c*x))/b)*cosh(5*a/b)/b^2/c^2+7/64*Chi(7*(a+b*arcsinh(c *x))/b)*cosh(7*a/b)/b^2/c^2-5/64*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c ^2-27/64*Shi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^2-25/64*Shi(5*(a+b* arcsinh(c*x))/b)*sinh(5*a/b)/b^2/c^2-7/64*Shi(7*(a+b*arcsinh(c*x))/b)*sinh (7*a/b)/b^2/c^2
Time = 0.87 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.47 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {64 b c x+192 b c^3 x^3+192 b c^5 x^5+64 b c^7 x^7-5 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-27 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 b \text {arcsinh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+27 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+27 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 b \text {arcsinh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b^2 c^2 (a+b \text {arcsinh}(c x))} \]
-1/64*(64*b*c*x + 192*b*c^3*x^3 + 192*b*c^5*x^5 + 64*b*c^7*x^7 - 5*(a + b* ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 27*(a + b*ArcSi nh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 25*a*Cosh[(5 *a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 25*b*ArcSinh[c*x]*Cosh[(5*a) /b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 7*a*Cosh[(7*a)/b]*CoshIntegral[ 7*(a/b + ArcSinh[c*x])] - 7*b*ArcSinh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a /b + ArcSinh[c*x])] + 5*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 5*b *ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 27*a*Sinh[(3*a) /b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 27*b*ArcSinh[c*x]*Sinh[(3*a)/b] *SinhIntegral[3*(a/b + ArcSinh[c*x])] + 25*a*Sinh[(5*a)/b]*SinhIntegral[5* (a/b + ArcSinh[c*x])] + 25*b*ArcSinh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/ b + ArcSinh[c*x])] + 7*a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x]) ] + 7*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/( b^2*c^2*(a + b*ArcSinh[c*x]))
Time = 1.62 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6229, 6206, 3042, 3793, 2009, 6234, 5971, 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.
| \(\displaystyle \int \frac {x \left (c^2 x^2+1\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6229 |
| \(\displaystyle \frac {\int \frac {\left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b c}+\frac {7 c \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 6206 |
| \(\displaystyle \frac {\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {7 c \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^5}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {7 c \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {7 c \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 c \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}+\frac {\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {7 \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {7 \int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {3 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}+\frac {7 \left (-\frac {5}{64} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {3}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {5}{64} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {1}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\) |
-((x*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) + ((5*Cosh[a/b]*CoshInte gral[(a + b*ArcSinh[c*x])/b])/8 + (5*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b* ArcSinh[c*x]))/b])/16 + (Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x] ))/b])/16 - (5*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/8 - (5*Sinh [(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/16 - (Sinh[(5*a)/b]*Si nhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/16)/(b^2*c^2) + (7*((-5*Cosh[a/b]* CoshIntegral[(a + b*ArcSinh[c*x])/b])/64 + (Cosh[(3*a)/b]*CoshIntegral[(3* (a + b*ArcSinh[c*x]))/b])/64 + (3*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*Arc Sinh[c*x]))/b])/64 + (Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/ b])/64 + (5*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/64 - (Sinh[(3* a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/64 - (3*Sinh[(5*a)/b]*Sinh Integral[(5*(a + b*ArcSinh[c*x]))/b])/64 - (Sinh[(7*a)/b]*SinhIntegral[(7* (a + b*ArcSinh[c*x]))/b])/64))/(b^2*c^2)
3.5.29.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Subst[Int [x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p *((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f* (n + 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] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(259)=518\).
Time = 0.28 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.48
-1/128*(64*c^7*x^7-64*c^6*x^6*(c^2*x^2+1)^(1/2)+112*c^5*x^5-80*c^4*x^4*(c^ 2*x^2+1)^(1/2)+56*c^3*x^3-24*c^2*x^2*(c^2*x^2+1)^(1/2)+7*c*x-(c^2*x^2+1)^( 1/2))/c^2/(a+b*arcsinh(c*x))/b-7/128/c^2/b^2*exp(7*a/b)*Ei(1,7*arcsinh(c*x )+7*a/b)-5/128*(16*c^5*x^5-16*c^4*x^4*(c^2*x^2+1)^(1/2)+20*c^3*x^3-12*c^2* x^2*(c^2*x^2+1)^(1/2)+5*c*x-(c^2*x^2+1)^(1/2))/c^2/b/(a+b*arcsinh(c*x))-25 /128/c^2/b^2*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)-9/128*(4*c^3*x^3-4*c^2* x^2*(c^2*x^2+1)^(1/2)+3*c*x-(c^2*x^2+1)^(1/2))/c^2/b/(a+b*arcsinh(c*x))-27 /128/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-5/128*(-(c^2*x^2+1)^(1/ 2)+c*x)/c^2/b/(a+b*arcsinh(c*x))-5/128/c^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+ a/b)-5/128/c^2/b^2*(arcsinh(c*x)*Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*b+Ei(1, -arcsinh(c*x)-a/b)*exp(-a/b)*a+b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x ))-9/128/c^2/b^2*(4*b*c^3*x^3+4*(c^2*x^2+1)^(1/2)*b*c^2*x^2+3*arcsinh(c*x) *Ei(1,-3*arcsinh(c*x)-3*a/b)*exp(-3*a/b)*b+3*Ei(1,-3*arcsinh(c*x)-3*a/b)*e xp(-3*a/b)*a+3*b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))-5/128/c^2/b^2 *(16*b*c^5*x^5+16*(c^2*x^2+1)^(1/2)*b*c^4*x^4+20*b*c^3*x^3+12*(c^2*x^2+1)^ (1/2)*b*c^2*x^2+5*arcsinh(c*x)*Ei(1,-5*arcsinh(c*x)-5*a/b)*exp(-5*a/b)*b+5 *Ei(1,-5*arcsinh(c*x)-5*a/b)*exp(-5*a/b)*a+5*b*c*x+(c^2*x^2+1)^(1/2)*b)/(a +b*arcsinh(c*x))-1/128/c^2/b^2*(64*b*c^7*x^7+64*(c^2*x^2+1)^(1/2)*b*c^6*x^ 6+112*b*c^5*x^5+80*(c^2*x^2+1)^(1/2)*b*c^4*x^4+56*b*c^3*x^3+24*(c^2*x^2+1) ^(1/2)*b*c^2*x^2+7*arcsinh(c*x)*Ei(1,-7*arcsinh(c*x)-7*a/b)*exp(-7*a/b)...
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
integral((c^4*x^5 + 2*c^2*x^3 + x)*sqrt(c^2*x^2 + 1)/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-((c^6*x^7 + 3*c^4*x^5 + 3*c^2*x^3 + x)*(c^2*x^2 + 1) + (c^7*x^8 + 3*c^5*x ^6 + 3*c^3*x^4 + c*x^2)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1 )*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)* log(c*x + sqrt(c^2*x^2 + 1))) + integrate((7*(c^7*x^7 + 2*c^5*x^5 + c^3*x^ 3)*(c^2*x^2 + 1)^(3/2) + (14*c^8*x^8 + 37*c^6*x^6 + 33*c^4*x^4 + 11*c^2*x^ 2 + 1)*(c^2*x^2 + 1) + (7*c^9*x^9 + 23*c^7*x^7 + 27*c^5*x^5 + 13*c^3*x^3 + 2*c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^4 + (c^2*x^2 + 1)*a*b*c^3*x^2 + 2*a* b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^2 + 2*b^2*c^3*x ^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2*c^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt (c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2*x^2 + 1)), x)
Exception generated. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]