\(\int (c+a^2 c x^2)^{5/2} \text {arcsinh}(a x)^3 \, dx\) [334]

Optimal result

Integrand size = 21, antiderivative size = 509 \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=-\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}} \]

[Out]

5/24*c*x*(a^2*c*x^2+c)^(3/2)*arcsinh(a*x)^3+1/6*x*(a^2*c*x^2+c)^(5/2)*arcsinh(a*x)^3-1/216*c^2*(a^2*x^2+1)^(5/
2)*(a^2*c*x^2+c)^(1/2)/a+245/384*c^2*x*arcsinh(a*x)*(a^2*c*x^2+c)^(1/2)+65/576*c^2*x*(a^2*x^2+1)*arcsinh(a*x)*
(a^2*c*x^2+c)^(1/2)+1/36*c^2*x*(a^2*x^2+1)^2*arcsinh(a*x)*(a^2*c*x^2+c)^(1/2)-5/32*c^2*(a^2*x^2+1)^(3/2)*arcsi
nh(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-1/12*c^2*(a^2*x^2+1)^(5/2)*arcsinh(a*x)^2*(a^2*c*x^2+c)^(1/2)/a+5/16*c^2*x*arc
sinh(a*x)^3*(a^2*c*x^2+c)^(1/2)-865/2304*a*c^2*x^2*(a^2*c*x^2+c)^(1/2)/(a^2*x^2+1)^(1/2)-65/2304*a^3*c^2*x^4*(
a^2*c*x^2+c)^(1/2)/(a^2*x^2+1)^(1/2)-115/768*c^2*arcsinh(a*x)^2*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)-15/32*
a*c^2*x^2*arcsinh(a*x)^2*(a^2*c*x^2+c)^(1/2)/(a^2*x^2+1)^(1/2)+5/64*c^2*arcsinh(a*x)^4*(a^2*c*x^2+c)^(1/2)/a/(
a^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5786, 5785, 5783, 5776, 5812, 30, 5798, 14, 267} \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=-\frac {15 a c^2 x^2 \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{32 \sqrt {a^2 x^2+1}}+\frac {5}{16} c^2 x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}+\frac {245}{384} c^2 x \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {1}{36} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {65}{576} c^2 x \left (a^2 x^2+1\right ) \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {5 c^2 \text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{64 a \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{12 a}-\frac {5 c^2 \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{32 a}-\frac {115 c^2 \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{768 a \sqrt {a^2 x^2+1}}+\frac {1}{6} x \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^{5/2}+\frac {5}{24} c x \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {865 a c^2 x^2 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt {a^2 c x^2+c}}{216 a}-\frac {65 a^3 c^2 x^4 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}} \]

[In]

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

[Out]

(-865*a*c^2*x^2*Sqrt[c + a^2*c*x^2])/(2304*Sqrt[1 + a^2*x^2]) - (65*a^3*c^2*x^4*Sqrt[c + a^2*c*x^2])/(2304*Sqr
t[1 + a^2*x^2]) - (c^2*(1 + a^2*x^2)^(5/2)*Sqrt[c + a^2*c*x^2])/(216*a) + (245*c^2*x*Sqrt[c + a^2*c*x^2]*ArcSi
nh[a*x])/384 + (65*c^2*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/576 + (c^2*x*(1 + a^2*x^2)^2*Sqrt[c +
a^2*c*x^2]*ArcSinh[a*x])/36 - (115*c^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(768*a*Sqrt[1 + a^2*x^2]) - (15*a*
c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(32*Sqrt[1 + a^2*x^2]) - (5*c^2*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c
*x^2]*ArcSinh[a*x]^2)/(32*a) - (c^2*(1 + a^2*x^2)^(5/2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(12*a) + (5*c^2*x*
Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^3)/24 + (x*(c + a^2*c*x^2)^
(5/2)*ArcSinh[a*x]^3)/6 + (5*c^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(64*a*Sqrt[1 + a^2*x^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5785

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcSinh[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(a + b*ArcSinh[c*x])^
n/Sqrt[1 + c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[x*(a + b*ArcSinh[c*x
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5786

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*(
(a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^
n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \text {arcsinh}(a x)^2 \, dx}{2 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{5/2} \text {arcsinh}(a x) \, dx}{6 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \text {arcsinh}(a x)^2 \, dx}{8 \sqrt {1+a^2 x^2}} \\ & = \frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x) \, dx}{36 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x) \, dx}{16 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \, dx}{36 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \text {arcsinh}(a x)^2 \, dx}{16 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \, dx}{48 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{64 \sqrt {1+a^2 x^2}}+\frac {\left (15 a^2 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{96 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{96 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{32 \sqrt {1+a^2 x^2}} \\ & = -\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.35 \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (4320 \text {arcsinh}(a x)^4-9720 \cosh (2 \text {arcsinh}(a x))-243 \cosh (4 \text {arcsinh}(a x))-8 \cosh (6 \text {arcsinh}(a x))-72 \text {arcsinh}(a x)^2 (270 \cosh (2 \text {arcsinh}(a x))+27 \cosh (4 \text {arcsinh}(a x))+2 \cosh (6 \text {arcsinh}(a x)))+288 \text {arcsinh}(a x)^3 (45 \sinh (2 \text {arcsinh}(a x))+9 \sinh (4 \text {arcsinh}(a x))+\sinh (6 \text {arcsinh}(a x)))+12 \text {arcsinh}(a x) (1620 \sinh (2 \text {arcsinh}(a x))+81 \sinh (4 \text {arcsinh}(a x))+4 \sinh (6 \text {arcsinh}(a x)))\right )}{55296 a \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(4320*ArcSinh[a*x]^4 - 9720*Cosh[2*ArcSinh[a*x]] - 243*Cosh[4*ArcSinh[a*x]] - 8*Cosh[
6*ArcSinh[a*x]] - 72*ArcSinh[a*x]^2*(270*Cosh[2*ArcSinh[a*x]] + 27*Cosh[4*ArcSinh[a*x]] + 2*Cosh[6*ArcSinh[a*x
]]) + 288*ArcSinh[a*x]^3*(45*Sinh[2*ArcSinh[a*x]] + 9*Sinh[4*ArcSinh[a*x]] + Sinh[6*ArcSinh[a*x]]) + 12*ArcSin
h[a*x]*(1620*Sinh[2*ArcSinh[a*x]] + 81*Sinh[4*ArcSinh[a*x]] + 4*Sinh[6*ArcSinh[a*x]])))/(55296*a*Sqrt[1 + a^2*
x^2])

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.58

method	result	size
default	\(\frac {5 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4} c^{2}}{64 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}+32 x^{6} a^{6} \sqrt {a^{2} x^{2}+1}+64 a^{5} x^{5}+48 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+38 a^{3} x^{3}+18 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+6 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (36 \operatorname {arcsinh}\left (a x \right )^{3}-18 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}+8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}+8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}-24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}-6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}-8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}+24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}-32 x^{6} a^{6} \sqrt {a^{2} x^{2}+1}+64 a^{5} x^{5}-48 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+38 a^{3} x^{3}-18 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+6 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (36 \operatorname {arcsinh}\left (a x \right )^{3}+18 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}\)	\(802\)

[In]

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

[Out]

5/64*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a*arcsinh(a*x)^4*c^2+1/13824*(c*(a^2*x^2+1))^(1/2)*(32*a^7*x^7+32
*x^6*a^6*(a^2*x^2+1)^(1/2)+64*a^5*x^5+48*a^4*x^4*(a^2*x^2+1)^(1/2)+38*a^3*x^3+18*a^2*x^2*(a^2*x^2+1)^(1/2)+6*a
*x+(a^2*x^2+1)^(1/2))*(36*arcsinh(a*x)^3-18*arcsinh(a*x)^2+6*arcsinh(a*x)-1)*c^2/a/(a^2*x^2+1)+3/4096*(c*(a^2*
x^2+1))^(1/2)*(8*a^5*x^5+8*a^4*x^4*(a^2*x^2+1)^(1/2)+12*a^3*x^3+8*a^2*x^2*(a^2*x^2+1)^(1/2)+4*a*x+(a^2*x^2+1)^
(1/2))*(32*arcsinh(a*x)^3-24*arcsinh(a*x)^2+12*arcsinh(a*x)-3)*c^2/a/(a^2*x^2+1)+15/512*(c*(a^2*x^2+1))^(1/2)*
(2*a^3*x^3+2*a^2*x^2*(a^2*x^2+1)^(1/2)+2*a*x+(a^2*x^2+1)^(1/2))*(4*arcsinh(a*x)^3-6*arcsinh(a*x)^2+6*arcsinh(a
*x)-3)*c^2/a/(a^2*x^2+1)+15/512*(c*(a^2*x^2+1))^(1/2)*(2*a^3*x^3-2*a^2*x^2*(a^2*x^2+1)^(1/2)+2*a*x-(a^2*x^2+1)
^(1/2))*(4*arcsinh(a*x)^3+6*arcsinh(a*x)^2+6*arcsinh(a*x)+3)*c^2/a/(a^2*x^2+1)+3/4096*(c*(a^2*x^2+1))^(1/2)*(8
*a^5*x^5-8*a^4*x^4*(a^2*x^2+1)^(1/2)+12*a^3*x^3-8*a^2*x^2*(a^2*x^2+1)^(1/2)+4*a*x-(a^2*x^2+1)^(1/2))*(32*arcsi
nh(a*x)^3+24*arcsinh(a*x)^2+12*arcsinh(a*x)+3)*c^2/a/(a^2*x^2+1)+1/13824*(c*(a^2*x^2+1))^(1/2)*(32*a^7*x^7-32*
x^6*a^6*(a^2*x^2+1)^(1/2)+64*a^5*x^5-48*a^4*x^4*(a^2*x^2+1)^(1/2)+38*a^3*x^3-18*a^2*x^2*(a^2*x^2+1)^(1/2)+6*a*
x-(a^2*x^2+1)^(1/2))*(36*arcsinh(a*x)^3+18*arcsinh(a*x)^2+6*arcsinh(a*x)+1)*c^2/a/(a^2*x^2+1)

Fricas [F]

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

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^3, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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