3.334 \(\int (c+a^2 c x^2)^{5/2} \sinh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=509 \[ -\frac{65 a^3 c^2 x^4 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}-\frac{865 a c^2 x^2 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}-\frac{c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt{a^2 c x^2+c}}{216 a}-\frac{15 a c^2 x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 \sqrt{a^2 x^2+1}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left (a^2 x^2+1\right )^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{65}{576} c^2 x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{64 a \sqrt{a^2 x^2+1}}-\frac{c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{12 a}-\frac{5 c^2 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 a}-\frac{115 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{768 a \sqrt{a^2 x^2+1}}+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^3 \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.594754, antiderivative size = 509, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5684, 5682, 5675, 5661, 5758, 30, 5717, 14, 261} \[ -\frac{65 a^3 c^2 x^4 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}-\frac{865 a c^2 x^2 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}-\frac{c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt{a^2 c x^2+c}}{216 a}-\frac{15 a c^2 x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 \sqrt{a^2 x^2+1}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left (a^2 x^2+1\right )^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{65}{576} c^2 x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{64 a \sqrt{a^2 x^2+1}}-\frac{c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{12 a}-\frac{5 c^2 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 a}-\frac{115 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{768 a \sqrt{a^2 x^2+1}}+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[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 5684

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1
+ c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 5682

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[Sqrt[d + e*x^2]/(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*Sqrt[d + e*x^2])/(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 5675

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

Rule 5661

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 5758

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

Rule 30

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

Rule 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 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 261

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]

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(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 \sinh ^{-1}(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} \sinh ^{-1}(a x)^2}{12 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \sinh ^{-1}(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} \sinh ^{-1}(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 ) \sinh ^{-1}(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} \sinh ^{-1}(a x)-\frac{5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac{c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(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} \sinh ^{-1}(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} \sinh ^{-1}(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{\sinh ^{-1}(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 \sinh ^{-1}(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} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)-\frac{15 a c^2 x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(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} \sinh ^{-1}(a x)^2}{32 a}-\frac{c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{5 c^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \sinh ^{-1}(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 \sinh ^{-1}(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} \sinh ^{-1}(a x)+\frac{65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)-\frac{15 a c^2 x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(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} \sinh ^{-1}(a x)^2}{32 a}-\frac{c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{5 c^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(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{\sinh ^{-1}(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{\sinh ^{-1}(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{\sinh ^{-1}(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} \sinh ^{-1}(a x)+\frac{65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)-\frac{115 c^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \sinh ^{-1}(a x)^2}{32 a}-\frac{c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac{5 c^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^4}{64 a \sqrt{1+a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.685039, size = 177, normalized size = 0.35 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (4320 \sinh ^{-1}(a x)^4+288 \left (45 \sinh \left (2 \sinh ^{-1}(a x)\right )+9 \sinh \left (4 \sinh ^{-1}(a x)\right )+\sinh \left (6 \sinh ^{-1}(a x)\right )\right ) \sinh ^{-1}(a x)^3+12 \left (1620 \sinh \left (2 \sinh ^{-1}(a x)\right )+81 \sinh \left (4 \sinh ^{-1}(a x)\right )+4 \sinh \left (6 \sinh ^{-1}(a x)\right )\right ) \sinh ^{-1}(a x)-72 \sinh ^{-1}(a x)^2 \left (270 \cosh \left (2 \sinh ^{-1}(a x)\right )+27 \cosh \left (4 \sinh ^{-1}(a x)\right )+2 \cosh \left (6 \sinh ^{-1}(a x)\right )\right )-9720 \cosh \left (2 \sinh ^{-1}(a x)\right )-243 \cosh \left (4 \sinh ^{-1}(a x)\right )-8 \cosh \left (6 \sinh ^{-1}(a x)\right )\right )}{55296 a \sqrt{a^2 x^2+1}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.176, size = 802, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*x^7*a^7+32
*a^6*x^6*(a^2*x^2+1)^(1/2)+64*x^5*a^5+48*a^4*x^4*(a^2*x^2+1)^(1/2)+38*x^3*a^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*x^5*a^5+8*a^4*x^4*(a^2*x^2+1)^(1/2)+12*x^3*a^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*x^3*a^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*x^3*a^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
*x^5*a^5-8*a^4*x^4*(a^2*x^2+1)^(1/2)+12*x^3*a^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*x^7*a^7-32*
a^6*x^6*(a^2*x^2+1)^(1/2)+64*x^5*a^5-48*a^4*x^4*(a^2*x^2+1)^(1/2)+38*x^3*a^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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out