3.5.19 \(\int \frac {x^3 (1+c^2 x^2)^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [419]

3.5.19.1 Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^3 \left (1+c^2 x^2\right )^2}{b c (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}-\frac {9 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^4} \]

output

-x^3*(c^2*x^2+1)^2/b/c/(a+b*arcsinh(c*x))-3/64*Chi((a+b*arcsinh(c*x))/b)*c 
osh(a/b)/b^2/c^4-9/64*Chi(3*(a+b*arcsinh(c*x))/b)*cosh(3*a/b)/b^2/c^4+5/64 
*Chi(5*(a+b*arcsinh(c*x))/b)*cosh(5*a/b)/b^2/c^4+7/64*Chi(7*(a+b*arcsinh(c 
*x))/b)*cosh(7*a/b)/b^2/c^4+3/64*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c 
^4+9/64*Shi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^4-5/64*Shi(5*(a+b*ar 
csinh(c*x))/b)*sinh(5*a/b)/b^2/c^4-7/64*Shi(7*(a+b*arcsinh(c*x))/b)*sinh(7 
*a/b)/b^2/c^4
 

3.5.19.2 Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.44 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {-64 b c^3 x^3-128 b c^5 x^5-64 b c^7 x^7-3 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-9 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 b \text {arcsinh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+3 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+9 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 b \text {arcsinh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b^2 c^4 (a+b \text {arcsinh}(c x))} \]

input

Integrate[(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]
 

output

(-64*b*c^3*x^3 - 128*b*c^5*x^5 - 64*b*c^7*x^7 - 3*(a + b*ArcSinh[c*x])*Cos 
h[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 9*(a + b*ArcSinh[c*x])*Cosh[(3*a 
)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + 5*a*Cosh[(5*a)/b]*CoshIntegral 
[5*(a/b + ArcSinh[c*x])] + 5*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])] 
+ 3*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*b*ArcSinh[c*x]*Sinh[a 
/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 9*a*Sinh[(3*a)/b]*SinhIntegral[3*(a 
/b + ArcSinh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + 
 ArcSinh[c*x])] - 5*a*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] - 
 5*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]*Sin 
h[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b^2*c^4*(a + b*ArcSin 
h[c*x]))
 

3.5.19.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.42, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6229, 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.

\(\displaystyle \int \frac {x^3 \left (c^2 x^2+1\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6229

\(\displaystyle \frac {3 \int \frac {x^2 \left (c^2 x^2+1\right )}{a+b \text {arcsinh}(c x)}dx}{b c}+\frac {7 c \int \frac {x^4 \left (c^2 x^2+1\right )}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x^3 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {7 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {3 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {3 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {7 \int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {1}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^4}+\frac {7 \left (\frac {3}{64} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {3}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{64} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {3}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^2}{b c (a+b \text {arcsinh}(c x))}\)

input

Int[(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]
 

output

-((x^3*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) + (3*(-1/8*(Cosh[a/b]* 
CoshIntegral[(a + b*ArcSinh[c*x])/b]) + (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 + (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/8 - (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^4) + (7*((3*Cosh[a/b]* 
CoshIntegral[(a + b*ArcSinh[c*x])/b])/64 - (3*Cosh[(3*a)/b]*CoshIntegral[( 
3*(a + b*ArcSinh[c*x]))/b])/64 - (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 - (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/64 + (3*Sinh[( 
3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/64 + (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^4)
 

3.5.19.3.1 Defintions of rubi rules used

rule 2009

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

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 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]
 

3.5.19.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(261)=522\).

Time = 0.31 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.46

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

input

int(x^3*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x,method=_RETURNVERBOSE)
 

output

-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^4/(a+b*arcsinh(c*x))/b-7/128/c^4/b^2*exp(7*a/b)*Ei(1,7*arcsinh(c*x 
)+7*a/b)-1/128*(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^4/b/(a+b*arcsinh(c*x))-5/ 
128/c^4/b^2*exp(5*a/b)*Ei(1,5*arcsinh(c*x)+5*a/b)+3/128*(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^4/b/(a+b*arcsinh(c*x))+9/1 
28/c^4/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)+3/128*(-(c^2*x^2+1)^(1/2) 
+c*x)/c^4/b/(a+b*arcsinh(c*x))+3/128/c^4/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/ 
b)+3/128/c^4/b^2*(arcsinh(c*x)*Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*b+Ei(1,-a 
rcsinh(c*x)-a/b)*exp(-a/b)*a+b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x)) 
+3/128/c^4/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)*E 
i(1,-3*arcsinh(c*x)-3*a/b)*exp(-3*a/b)*b+3*Ei(1,-3*arcsinh(c*x)-3*a/b)*exp 
(-3*a/b)*a+3*b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))-1/128/c^4/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*E 
i(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^4/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)*b...
 

3.5.19.5 Fricas [F]

\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^3*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas" 
)
 

output

integral((c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)/(b^2*arcsinh(c*x)^2 + 2*a*b*arc 
sinh(c*x) + a^2), x)
 

3.5.19.6 Sympy [F]

\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{3} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

input

integrate(x**3*(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x))**2,x)
 

output

Integral(x**3*(c**2*x**2 + 1)**(3/2)/(a + b*asinh(c*x))**2, x)
 

3.5.19.7 Maxima [F]

\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^3*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima" 
)
 

output

-((c^4*x^7 + 2*c^2*x^5 + x^3)*(c^2*x^2 + 1) + (c^5*x^8 + 2*c^3*x^6 + c*x^4 
)*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^5*x^7 + 9*c^3*x^5 + 2*c*x^3)*(c^2*x^2 + 1)^(3/2) 
 + (14*c^6*x^8 + 27*c^4*x^6 + 16*c^2*x^4 + 3*x^2)*(c^2*x^2 + 1) + (7*c^7*x 
^9 + 18*c^5*x^7 + 15*c^3*x^5 + 4*c*x^3)*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)*s 
qrt(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)
 

3.5.19.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(x^3*(c^2*x^2+1)^(3/2)/(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
 

3.5.19.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^3\,{\left (c^2\,x^2+1\right )}^{3/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

input

int((x^3*(c^2*x^2 + 1)^(3/2))/(a + b*asinh(c*x))^2,x)
 

output

int((x^3*(c^2*x^2 + 1)^(3/2))/(a + b*asinh(c*x))^2, x)