\(\int \frac {x^3 (1+c^2 x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [412]

Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 b^2 c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b^2 c^4}+\frac {21 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}+\frac {9 \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4} \] Output:

-x^3*(c^2*x^2+1)^3/b/c/(a+b*arcsinh(c*x))-3/128*cosh(a/b)*Chi((a+b*arcsinh 
(c*x))/b)/b^2/c^4-3/32*cosh(3*a/b)*Chi(3*(a+b*arcsinh(c*x))/b)/b^2/c^4+21/ 
256*cosh(7*a/b)*Chi(7*(a+b*arcsinh(c*x))/b)/b^2/c^4+9/256*cosh(9*a/b)*Chi( 
9*(a+b*arcsinh(c*x))/b)/b^2/c^4+3/128*sinh(a/b)*Shi((a+b*arcsinh(c*x))/b)/ 
b^2/c^4+3/32*sinh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^4-21/256*sinh(7 
*a/b)*Shi(7*(a+b*arcsinh(c*x))/b)/b^2/c^4-9/256*sinh(9*a/b)*Shi(9*(a+b*arc 
sinh(c*x))/b)/b^2/c^4
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {256 b c^3 x^3+768 b c^5 x^5+768 b c^7 x^7+256 b c^9 x^9+6 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+24 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-21 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-21 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 a \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 b \text {arcsinh}(c x) \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-6 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-6 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-24 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+21 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+21 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 a \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 b \text {arcsinh}(c x) \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{256 b^2 c^4 (a+b \text {arcsinh}(c x))} \] Input:

Integrate[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-1/256*(256*b*c^3*x^3 + 768*b*c^5*x^5 + 768*b*c^7*x^7 + 256*b*c^9*x^9 + 6* 
(a + b*ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + 24*(a + 
b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 21*a* 
Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 21*b*ArcSinh[c*x]*Cos 
h[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 9*a*Cosh[(9*a)/b]*CoshIn 
tegral[9*(a/b + ArcSinh[c*x])] - 9*b*ArcSinh[c*x]*Cosh[(9*a)/b]*CoshIntegr 
al[9*(a/b + ArcSinh[c*x])] - 6*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x] 
] - 6*b*ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 24*a*Sin 
h[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 24*b*ArcSinh[c*x]*Sinh[( 
3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 21*a*Sinh[(7*a)/b]*SinhInte 
gral[7*(a/b + ArcSinh[c*x])] + 21*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegra 
l[7*(a/b + ArcSinh[c*x])] + 9*a*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSin 
h[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c* 
x])])/(b^2*c^4*(a + b*ArcSinh[c*x]))
 

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.78, 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.

\(\displaystyle \int \frac {x^3 \left (c^2 x^2+1\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6229

\(\displaystyle \frac {3 \int \frac {x^2 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b c}+\frac {9 c \int \frac {x^4 \left (c^2 x^2+1\right )^2}{a+b \text {arcsinh}(c x)}dx}{b}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {9 \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {3 \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {3 \int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {3 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}+\frac {9 \int \left (\frac {\cosh \left (\frac {9 a}{b}-\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \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^4}+\frac {9 \left (\frac {3}{128} \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 {1}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{256} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{256} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {3}{128} \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 {1}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{256} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{256} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))}\)

Input:

Int[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]
 

Output:

-((x^3*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) + (3*((-5*Cosh[a/b]*Co 
shIntegral[(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*ArcSi 
nh[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]*SinhIn 
tegral[(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) + (9*((3*Cosh[a/b]*CoshIntegral[(a 
+ b*ArcSinh[c*x])/b])/128 - (Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[ 
c*x]))/b])/64 - (Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/6 
4 + (Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/256 + (Cosh[( 
9*a)/b]*CoshIntegral[(9*(a + b*ArcSinh[c*x]))/b])/256 - (3*Sinh[a/b]*SinhI 
ntegral[(a + b*ArcSinh[c*x])/b])/128 + (Sinh[(3*a)/b]*SinhIntegral[(3*(a + 
 b*ArcSinh[c*x]))/b])/64 + (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c 
*x]))/b])/64 - (Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/25 
6 - (Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/256))/(b^2*c^ 
4)
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1069\) vs. \(2(261)=522\).

Time = 2.69 (sec) , antiderivative size = 1070, normalized size of antiderivative = 3.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1070\)

Input:

int(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/512*(256*c^9*x^9-256*(c^2*x^2+1)^(1/2)*x^8*c^8+576*x^7*c^7-448*x^6*c^6* 
(c^2*x^2+1)^(1/2)+432*x^5*c^5-240*x^4*c^4*(c^2*x^2+1)^(1/2)+120*x^3*c^3-40 
*x^2*c^2*(c^2*x^2+1)^(1/2)+9*x*c-(c^2*x^2+1)^(1/2))/c^4/b/(a+b*arcsinh(x*c 
))-9/512/c^4/b^2*exp(9*a/b)*Ei(1,9*arcsinh(x*c)+9*a/b)-3/512*(64*x^7*c^7-6 
4*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^4/b/(a+b*arc 
sinh(x*c))-21/512/c^4/b^2*exp(7*a/b)*Ei(1,7*arcsinh(x*c)+7*a/b)+1/64*(-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^4/b/(a+b*arc 
sinh(x*c))+3/64/c^4/b^2*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)+3/256*(x*c-( 
c^2*x^2+1)^(1/2))/c^4/b/(a+b*arcsinh(x*c))+3/256/c^4/b^2*exp(a/b)*Ei(1,arc 
sinh(x*c)+a/b)+3/256/c^4/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*a 
rcsinh(x*c))+1/64/c^4/b^2*(4*b*c^3*x^3+4*(c^2*x^2+1)^(1/2)*b*c^2*x^2+3*arc 
sinh(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))-3/51 
2/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*ex 
p(-7*a/b)*Ei(1,-7*arcsinh(x*c)-7*a/b)*b*arcsinh(x*c)+7*exp(-7*a/b)*Ei(1,-7 
*arcsinh(x*c)-7*a/b)*a+7*b*x*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(x*c))-1/5 
12/c^4/b^2*(256*b*c^9*x^9+256*(c^2*x^2+1)^(1/2)*b*c^8*x^8+576*b*c^7*x^7...
 

Fricas [F]

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

integrate(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas" 
)
 

Output:

integral((c^4*x^7 + 2*c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)/(b^2*arcsinh(c*x)^2 
 + 2*a*b*arcsinh(c*x) + a^2), x)
 

Sympy [F]

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

integrate(x**3*(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)
 

Output:

Integral(x**3*(c**2*x**2 + 1)**(5/2)/(a + b*asinh(c*x))**2, x)
 

Maxima [F]

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

integrate(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima" 
)
 

Output:

-((c^6*x^9 + 3*c^4*x^7 + 3*c^2*x^5 + x^3)*(c^2*x^2 + 1) + (c^7*x^10 + 3*c^ 
5*x^8 + 3*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(((9*c^7*x^9 + 20*c^5*x^7 + 13 
*c^3*x^5 + 2*c*x^3)*(c^2*x^2 + 1)^(3/2) + 3*(6*c^8*x^10 + 17*c^6*x^8 + 17* 
c^4*x^6 + 7*c^2*x^4 + x^2)*(c^2*x^2 + 1) + (9*c^9*x^11 + 31*c^7*x^9 + 39*c 
^5*x^7 + 21*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)*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. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

int((x^3*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2,x)
 

Output:

int((x^3*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{7}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}+2 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{5}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{3}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \] Input:

int(x^3*(c^2*x^2+1)^(5/2)/(a+b*asinh(c*x))^2,x)
 

Output:

int((sqrt(c**2*x**2 + 1)*x**7)/(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**5)/(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**3)/(asinh(c*x)** 
2*b**2 + 2*asinh(c*x)*a*b + a**2),x)