\(\int \frac {x (1+c^2 x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [414]

Optimal result

Integrand size = 25, antiderivative size = 275 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2} \] Output:

-x*(c^2*x^2+1)^3/b/c/(a+b*arcsinh(c*x))+5/64*cosh(a/b)*Chi((a+b*arcsinh(c* 
x))/b)/b^2/c^2+27/64*cosh(3*a/b)*Chi(3*(a+b*arcsinh(c*x))/b)/b^2/c^2+25/64 
*cosh(5*a/b)*Chi(5*(a+b*arcsinh(c*x))/b)/b^2/c^2+7/64*cosh(7*a/b)*Chi(7*(a 
+b*arcsinh(c*x))/b)/b^2/c^2-5/64*sinh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c 
^2-27/64*sinh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^2-25/64*sinh(5*a/b) 
*Shi(5*(a+b*arcsinh(c*x))/b)/b^2/c^2-7/64*sinh(7*a/b)*Shi(7*(a+b*arcsinh(c 
*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.47 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {64 b c x+192 b c^3 x^3+192 b c^5 x^5+64 b c^7 x^7-5 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-27 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 b \text {arcsinh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+27 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+27 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 b \text {arcsinh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b^2 c^2 (a+b \text {arcsinh}(c x))} \] Input:

Integrate[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-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]))
 

Rubi [A] (verified)

Time = 1.68 (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

\(\displaystyle \frac {\frac {5}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {5}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {5}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}+\frac {7 \left (-\frac {5}{64} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {3}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {5}{64} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {1}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\)

Input:

Int[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-((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)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

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]))
 

rule 5971

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]
 

rule 6206

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]
 

rule 6229

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]
 

rule 6234

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(259)=518\).

Time = 2.60 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.48

method	result	size
default	\(\text {Expression too large to display}\)	\(958\)

Input:

int(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/128*(64*x^7*c^7-64*x^6*c^6*(c^2*x^2+1)^(1/2)+112*x^5*c^5-80*x^4*c^4*(c^ 
2*x^2+1)^(1/2)+56*x^3*c^3-24*x^2*c^2*(c^2*x^2+1)^(1/2)+7*x*c-(c^2*x^2+1)^( 
1/2))/c^2/b/(a+b*arcsinh(x*c))-7/128/c^2/b^2*exp(7*a/b)*Ei(1,7*arcsinh(x*c 
)+7*a/b)-5/128*(16*x^5*c^5-16*x^4*c^4*(c^2*x^2+1)^(1/2)+20*x^3*c^3-12*x^2* 
c^2*(c^2*x^2+1)^(1/2)+5*x*c-(c^2*x^2+1)^(1/2))/c^2/b/(a+b*arcsinh(x*c))-25 
/128/c^2/b^2*exp(5*a/b)*Ei(1,5*arcsinh(x*c)+5*a/b)-9/128*(-4*x^2*c^2*(c^2* 
x^2+1)^(1/2)+4*x^3*c^3-(c^2*x^2+1)^(1/2)+3*x*c)/c^2/b/(a+b*arcsinh(x*c))-2 
7/128/c^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)-5/128*(x*c-(c^2*x^2+1) 
^(1/2))/c^2/b/(a+b*arcsinh(x*c))-5/128/c^2/b^2*exp(a/b)*Ei(1,arcsinh(x*c)+ 
a/b)-5/128/c^2/b^2*(arcsinh(x*c)*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)*b+exp(- 
a/b)*Ei(1,-arcsinh(x*c)-a/b)*a+b*x*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(x*c 
))-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(x*c) 
*exp(-3*a/b)*Ei(1,-3*arcsinh(x*c)-3*a/b)*b+3*exp(-3*a/b)*Ei(1,-3*arcsinh(x 
*c)-3*a/b)*a+3*b*x*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(x*c))-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(x*c)*Ei(1,-5*arcsinh(x*c)-5*a/b)*exp(-5*a/b)*b+5 
*Ei(1,-5*arcsinh(x*c)-5*a/b)*exp(-5*a/b)*a+5*b*x*c+(c^2*x^2+1)^(1/2)*b)/(a 
+b*arcsinh(x*c))-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*exp(-7*a/b)*Ei(1,-7*arcsinh(x*c)-7*a/b)*b*arcsinh(x*...
 

Fricas [F]

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

integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

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

integrate(x*(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)
 

Output:

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

Maxima [F]

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

integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
 

Output:

-((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)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2,x)
 

Output:

int((x*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2, x)
 

Reduce [F]

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

int(x*(c^2*x^2+1)^(5/2)/(a+b*asinh(c*x))^2,x)
 

Output:

int((sqrt(c**2*x**2 + 1)*x**5)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a* 
*2),x)*c**4 + 2*int((sqrt(c**2*x**2 + 1)*x**3)/(asinh(c*x)**2*b**2 + 2*asi 
nh(c*x)*a*b + a**2),x)*c**2 + int((sqrt(c**2*x**2 + 1)*x)/(asinh(c*x)**2*b 
**2 + 2*asinh(c*x)*a*b + a**2),x)