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

Optimal. Leaf size=625 \[ \frac{4 a b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 x^9 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{c^2 x^2+1}}-\frac{38 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{c^2 x^2+1}}-\frac{2 b c d^2 x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{c^2 x^2+1}}+\frac{1}{21} d^2 x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{c^2 x^2+1}}+\frac{d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}-\frac{2 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2 b^2 d^2 \left (c^2 x^2+1\right )^4 \sqrt{c^2 d x^2+d}}{729 c^4}-\frac{50 b^2 d^2 \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d}}{27783 c^4}-\frac{160 b^2 d^2 \sqrt{c^2 d x^2+d}}{3969 c^4}-\frac{4 b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{1323 c^4}-\frac{80 b^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{11907 c^4}+\frac{4 b^2 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{63 c^3 \sqrt{c^2 x^2+1}} \]

[Out]

(-160*b^2*d^2*Sqrt[d + c^2*d*x^2])/(3969*c^4) + (4*a*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) -
(80*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(11907*c^4) - (4*b^2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/
(1323*c^4) - (50*b^2*d^2*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(27783*c^4) + (2*b^2*d^2*(1 + c^2*x^2)^4*Sqrt[d
+ c^2*d*x^2])/(729*c^4) + (4*b^2*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(63*c^3*Sqrt[1 + c^2*x^2]) - (2*b*d^2
*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(189*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^5*Sqrt[d + c^2*d*x^2]*
(a + b*ArcSinh[c*x]))/(21*Sqrt[1 + c^2*x^2]) - (38*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(44
1*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(81*Sqrt[1 + c^2*x^2]) - (2*
d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(63*c^4) + (d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2
)/(63*c^2) + (d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/21 + (5*d*x^4*(d + c^2*d*x^2)^(3/2)*(a + b*A
rcSinh[c*x])^2)/63 + (x^4*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/9

________________________________________________________________________________________

Rubi [A]  time = 1.23754, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 18, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5744, 5742, 5758, 5717, 5653, 261, 5661, 266, 43, 14, 5730, 12, 446, 77, 270, 1251, 897, 1153} \[ \frac{4 a b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 x^9 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{c^2 x^2+1}}-\frac{38 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{c^2 x^2+1}}-\frac{2 b c d^2 x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{c^2 x^2+1}}+\frac{1}{21} d^2 x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{c^2 x^2+1}}+\frac{d^2 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}-\frac{2 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{1}{9} x^4 \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2 b^2 d^2 \left (c^2 x^2+1\right )^4 \sqrt{c^2 d x^2+d}}{729 c^4}-\frac{50 b^2 d^2 \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d}}{27783 c^4}-\frac{160 b^2 d^2 \sqrt{c^2 d x^2+d}}{3969 c^4}-\frac{4 b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{1323 c^4}-\frac{80 b^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{11907 c^4}+\frac{4 b^2 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{63 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-160*b^2*d^2*Sqrt[d + c^2*d*x^2])/(3969*c^4) + (4*a*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) -
(80*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(11907*c^4) - (4*b^2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/
(1323*c^4) - (50*b^2*d^2*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(27783*c^4) + (2*b^2*d^2*(1 + c^2*x^2)^4*Sqrt[d
+ c^2*d*x^2])/(729*c^4) + (4*b^2*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(63*c^3*Sqrt[1 + c^2*x^2]) - (2*b*d^2
*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(189*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^5*Sqrt[d + c^2*d*x^2]*
(a + b*ArcSinh[c*x]))/(21*Sqrt[1 + c^2*x^2]) - (38*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(44
1*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(81*Sqrt[1 + c^2*x^2]) - (2*
d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(63*c^4) + (d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2
)/(63*c^2) + (d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/21 + (5*d*x^4*(d + c^2*d*x^2)^(3/2)*(a + b*A
rcSinh[c*x])^2)/63 + (x^4*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/9

Rule 5744

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

Rule 5742

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
(f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f
, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 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 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 5653

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

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]

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 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 5730

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} (5 d) \int x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (2 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{9 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt{1+c^2 x^2}}-\frac{4 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{63 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{21} \left (5 d^2\right ) \int x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (10 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{63 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt{1+c^2 x^2}} \, dx}{9 \sqrt{1+c^2 x^2}}\\ &=-\frac{8 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{21 \sqrt{1+c^2 x^2}}-\frac{\left (2 b c d^2 \sqrt{d+c^2 d x^2}\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{21 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{1+c^2 x^2}} \, dx}{2835 \sqrt{1+c^2 x^2}}+\frac{\left (10 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt{1+c^2 x^2}} \, dx}{63 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}+\frac{d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{63 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (2 b d^2 \sqrt{d+c^2 d x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{63 c \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (63+90 c^2 x+35 c^4 x^2\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{2835 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (7+5 c^2 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx}{441 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{1+c^2 x^2}} \, dx}{105 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{1+c^2 x^2}}-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}-\frac{2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt{1+c^2 x^2}\right )}{2835 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2}} \, dx}{189 \sqrt{1+c^2 x^2}}+\frac{\left (4 b d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (7+5 c^2 x\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{441 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{105 \sqrt{1+c^2 x^2}}\\ &=\frac{4 a b d^2 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{1+c^2 x^2}}-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}-\frac{2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^4}+\frac{4 x^2}{c^4}+\frac{3 x^4}{c^4}-\frac{50 x^6}{c^4}+\frac{35 x^8}{c^4}\right ) \, dx,x,\sqrt{1+c^2 x^2}\right )}{2835 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{189 \sqrt{1+c^2 x^2}}+\frac{\left (4 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{c^4 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^4}-\frac{8 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1+c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{441 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4 \sqrt{1+c^2 x}}-\frac{2 \sqrt{1+c^2 x}}{c^4}+\frac{\left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{105 \sqrt{1+c^2 x^2}}\\ &=\frac{134 b^2 d^2 \sqrt{d+c^2 d x^2}}{3969 c^4}+\frac{4 a b d^2 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{122 b^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{11907 c^4}-\frac{4 b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}}{1323 c^4}-\frac{50 b^2 d^2 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2}}{27783 c^4}+\frac{2 b^2 d^2 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2}}{729 c^4}+\frac{4 b^2 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{1+c^2 x^2}}-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}-\frac{2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{189 \sqrt{1+c^2 x^2}}-\frac{\left (4 b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{63 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{160 b^2 d^2 \sqrt{d+c^2 d x^2}}{3969 c^4}+\frac{4 a b d^2 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{80 b^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{11907 c^4}-\frac{4 b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}}{1323 c^4}-\frac{50 b^2 d^2 \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2}}{27783 c^4}+\frac{2 b^2 d^2 \left (1+c^2 x^2\right )^4 \sqrt{d+c^2 d x^2}}{729 c^4}+\frac{4 b^2 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{189 c \sqrt{1+c^2 x^2}}-\frac{2 b c d^2 x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{21 \sqrt{1+c^2 x^2}}-\frac{38 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{441 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^9 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{81 \sqrt{1+c^2 x^2}}-\frac{2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^4}+\frac{d^2 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{63 c^2}+\frac{1}{21} d^2 x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{63} d x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} x^4 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.446444, size = 277, normalized size = 0.44 \[ \frac{d^2 \sqrt{c^2 d x^2+d} \left (3969 a^2 \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^4-126 a b c x \left (49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right ) \sqrt{c^2 x^2+1}-126 b \sinh ^{-1}(c x) \left (b c x \sqrt{c^2 x^2+1} \left (49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right )-63 a \left (c^2 x^2+1\right )^4 \left (7 c^2 x^2-2\right )\right )+2 b^2 \left (343 c^{10} x^{10}+1490 c^8 x^8+2152 c^6 x^6+106 c^4 x^4-7039 c^2 x^2-6140\right )+3969 b^2 \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^4 \sinh ^{-1}(c x)^2\right )}{250047 c^4 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*Sqrt[d + c^2*d*x^2]*(3969*a^2*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - 126*a*b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21
*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(-6140 - 7039*c^2*x^2 + 106*c^4*x^4 + 2152*c^6*x^6
+ 1490*c^8*x^8 + 343*c^10*x^10) - 126*b*(-63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(-12
6 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8))*ArcSinh[c*x] + 3969*b^2*(1 + c^2*x^2)^4*(-2 + 7*c^2*
x^2)*ArcSinh[c*x]^2))/(250047*c^4*(1 + c^2*x^2))

________________________________________________________________________________________

Maple [B]  time = 0.388, size = 2014, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(1/9*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(c^2*d*x^2+d)^(7/2))+b^2*(1/373248*(d*(c^2*x^2+1))^(1/2)*(25
6*c^10*x^10+256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8+576*c^7*x^7*(c^2*x^2+1)^(1/2)+688*c^6*x^6+432*c^5*x^5*(c
^2*x^2+1)^(1/2)+280*c^4*x^4+120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2+9*c*x*(c^2*x^2+1)^(1/2)+1)*(81*arcsinh(c*
x)^2-18*arcsinh(c*x)+2)*d^2/c^4/(c^2*x^2+1)+3/175616*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^
(1/2)+144*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2
*x^2+1)^(1/2)+1)*(49*arcsinh(c*x)^2-14*arcsinh(c*x)+2)*d^2/c^4/(c^2*x^2+1)-1/1728*(d*(c^2*x^2+1))^(1/2)*(4*c^4
*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x)+2)*d^2/
c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2
)*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2+2*arcsinh(
c*x)+2)*d^2/c^4/(c^2*x^2+1)-1/1728*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*
x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2+6*arcsinh(c*x)+2)*d^2/c^4/(c^2*x^2+1)+3/175616*(d*(c^2*x^2+1))^(1/2)*
(64*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-56*c^3*x^3*(c^2
*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*(49*arcsinh(c*x)^2+14*arcsinh(c*x)+2)*d^2/c^4/(c^2*x^2+1)+
1/373248*(d*(c^2*x^2+1))^(1/2)*(256*c^10*x^10-256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8-576*c^7*x^7*(c^2*x^2+1
)^(1/2)+688*c^6*x^6-432*c^5*x^5*(c^2*x^2+1)^(1/2)+280*c^4*x^4-120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2-9*c*x*(
c^2*x^2+1)^(1/2)+1)*(81*arcsinh(c*x)^2+18*arcsinh(c*x)+2)*d^2/c^4/(c^2*x^2+1))+2*a*b*(1/41472*(d*(c^2*x^2+1))^
(1/2)*(256*c^10*x^10+256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8+576*c^7*x^7*(c^2*x^2+1)^(1/2)+688*c^6*x^6+432*c
^5*x^5*(c^2*x^2+1)^(1/2)+280*c^4*x^4+120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2+9*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+9
*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*
c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1
/2)+1)*(-1+7*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1
/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(
c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*
x*(c^2*x^2+1)^(1/2)+1)*(1+arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(
c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x
^2+1))^(1/2)*(64*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-56
*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*(1+7*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+1/4147
2*(d*(c^2*x^2+1))^(1/2)*(256*c^10*x^10-256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8-576*c^7*x^7*(c^2*x^2+1)^(1/2)
+688*c^6*x^6-432*c^5*x^5*(c^2*x^2+1)^(1/2)+280*c^4*x^4-120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2-9*c*x*(c^2*x^2
+1)^(1/2)+1)*(1+9*arcsinh(c*x))*d^2/c^4/(c^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(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.20798, size = 1181, normalized size = 1.89 \begin{align*} \frac{3969 \,{\left (7 \, b^{2} c^{10} d^{2} x^{10} + 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} + 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} - 2 \, b^{2} d^{2}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 126 \,{\left (441 \, a b c^{10} d^{2} x^{10} + 1638 \, a b c^{8} d^{2} x^{8} + 2142 \, a b c^{6} d^{2} x^{6} + 1008 \, a b c^{4} d^{2} x^{4} - 63 \, a b c^{2} d^{2} x^{2} - 126 \, a b d^{2} -{\left (49 \, b^{2} c^{9} d^{2} x^{9} + 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} + 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (343 \,{\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} + 2 \,{\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \,{\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} + 4 \,{\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} -{\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} - 2 \,{\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2} - 126 \,{\left (49 \, a b c^{9} d^{2} x^{9} + 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} + 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{250047 \,{\left (c^{6} x^{2} + c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/250047*(3969*(7*b^2*c^10*d^2*x^10 + 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2*x^6 + 16*b^2*c^4*d^2*x^4 - b^2*c^2*d
^2*x^2 - 2*b^2*d^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 126*(441*a*b*c^10*d^2*x^10 + 1638*a*b
*c^8*d^2*x^8 + 2142*a*b*c^6*d^2*x^6 + 1008*a*b*c^4*d^2*x^4 - 63*a*b*c^2*d^2*x^2 - 126*a*b*d^2 - (49*b^2*c^9*d^
2*x^9 + 171*b^2*c^7*d^2*x^7 + 189*b^2*c^5*d^2*x^5 + 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*s
qrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (343*(81*a^2 + 2*b^2)*c^10*d^2*x^10 + 2*(51597*a^2 + 1490*b^
2)*c^8*d^2*x^8 + 2*(67473*a^2 + 2152*b^2)*c^6*d^2*x^6 + 4*(15876*a^2 + 53*b^2)*c^4*d^2*x^4 - (3969*a^2 + 14078
*b^2)*c^2*d^2*x^2 - 2*(3969*a^2 + 6140*b^2)*d^2 - 126*(49*a*b*c^9*d^2*x^9 + 171*a*b*c^7*d^2*x^7 + 189*a*b*c^5*
d^2*x^5 + 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError