\(\int \frac {x^3 (1+c^2 x^2)^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx\) [427]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5814, 5819, 5556, 3384, 3379, 3382} \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 b^2 c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b^2 c^4}+\frac {21 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}+\frac {9 \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arcsinh}(c x))}{b}\right )}{256 b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))} \]

[In]

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

[Out]

-((x^3*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(128*
b^2*c^4) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) + (21*Cosh[(7*a)/b]*CoshInt
egral[(7*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4) + (9*Cosh[(9*a)/b]*CoshIntegral[(9*(a + b*ArcSinh[c*x]))/b])/
(256*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(128*b^2*c^4) + (3*Sinh[(3*a)/b]*SinhIntegr
al[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) - (21*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(25
6*b^2*c^4) - (9*Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

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 5814

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] + (-Dist[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] - Dist[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 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \int \frac {x^2 \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b c}+\frac {(9 c) \int \frac {x^4 \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4}+\frac {9 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}+\frac {3 \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4}+\frac {9 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{256 x}+\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{256 x}-\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}-\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}+\frac {3 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{128 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {9 \text {Subst}\left (\int \frac {\cosh \left (\frac {9 a}{b}-\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4}+\frac {9 \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4}+\frac {3 \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {3 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {9 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {27 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b^2 c^4}-\frac {15 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4} \\ & = -\frac {x^3 \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\left (27 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b^2 c^4}-\frac {\left (15 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {\left (9 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {\left (9 \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4}+\frac {\left (3 \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {\left (9 \cosh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4}-\frac {\left (27 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b^2 c^4}+\frac {\left (15 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}+\frac {\left (9 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {\left (9 \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4}-\frac {\left (3 \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^4}-\frac {\left (9 \sinh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {9 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b^2 c^4} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {256 b c^3 x^3+768 b c^5 x^5+768 b c^7 x^7+256 b c^9 x^9+6 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+24 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-21 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-21 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 a \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 b \text {arcsinh}(c x) \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-6 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-6 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-24 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+21 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+21 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 a \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+9 b \text {arcsinh}(c x) \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{256 b^2 c^4 (a+b \text {arcsinh}(c x))} \]

[In]

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

[Out]

-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]*CoshI
ntegral[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]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + A
rcSinh[c*x])] - 9*a*Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcSinh[c*x])] - 9*b*ArcSinh[c*x]*Cosh[(9*a)/b]*CoshIn
tegral[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]*S
inhIntegral[a/b + ArcSinh[c*x]] - 24*a*Sinh[(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]*SinhIntegral[7*(a/b + ArcSinh[c*x])] +
21*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 9*a*Sinh[(9*a)/b]*SinhIntegral[9*(a/b
+ ArcSinh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])])/(b^2*c^4*(a + b*ArcSin
h[c*x]))

Maple [B] (verified)

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

Time = 0.34 (sec) , antiderivative size = 1070, normalized size of antiderivative = 3.86

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-((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. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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