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

Optimal. Leaf size=405 \[ -\frac{b c^3 d x^6 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{c^2 x^2+1}}-\frac{7 b c d x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{c^2 x^2+1}}+\frac{1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{c^2 x^2+1}}+\frac{d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}-\frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{c^2 d x^2+d}+\frac{43 b^2 d x^3 \sqrt{c^2 d x^2+d}}{1728}-\frac{7 b^2 d x \sqrt{c^2 d x^2+d}}{1152 c^2}+\frac{7 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt{c^2 x^2+1}} \]

[Out]

(-7*b^2*d*x*Sqrt[d + c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d + c^2*d*x^2])/1728 + (b^2*c^2*d*x^5*Sqrt[d
+ c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(1152*c^3*Sqrt[1 + c^2*x^2]) - (b*d*x^2*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c*Sqrt[1 + c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x]))/(48*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(18*Sqrt[1 + c^2*x^2])
+ (d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*c^2) + (d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^
2)/8 + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/6 - (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(
48*b*c^3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.663907, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5744, 5742, 5758, 5675, 5661, 321, 215, 14, 5730, 12, 459} \[ -\frac{b c^3 d x^6 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{c^2 x^2+1}}-\frac{7 b c d x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{c^2 x^2+1}}+\frac{1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{8} d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{c^2 x^2+1}}+\frac{d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}-\frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{c^2 d x^2+d}+\frac{43 b^2 d x^3 \sqrt{c^2 d x^2+d}}{1728}-\frac{7 b^2 d x \sqrt{c^2 d x^2+d}}{1152 c^2}+\frac{7 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b^2*d*x*Sqrt[d + c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d + c^2*d*x^2])/1728 + (b^2*c^2*d*x^5*Sqrt[d
+ c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(1152*c^3*Sqrt[1 + c^2*x^2]) - (b*d*x^2*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c*Sqrt[1 + c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x]))/(48*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(18*Sqrt[1 + c^2*x^2])
+ (d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*c^2) + (d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^
2)/8 + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/6 - (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(
48*b*c^3*Sqrt[1 + c^2*x^2])

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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 459

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

Rubi steps

\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d \int x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}-\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (3+2 c^2 x^2\right )}{12 \sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{7 b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{16 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (b d \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 c \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4 \left (3+2 c^2 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx}{36 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{64} b^2 d x^3 \sqrt{d+c^2 d x^2}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{d+c^2 d x^2}-\frac{b d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{1+c^2 x^2}}-\frac{7 b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1+c^2 x^2}}-\frac{\left (3 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{64 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{27 \sqrt{1+c^2 x^2}}\\ &=\frac{b^2 d x \sqrt{d+c^2 d x^2}}{128 c^2}+\frac{43 b^2 d x^3 \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{d+c^2 d x^2}-\frac{b d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{1+c^2 x^2}}-\frac{7 b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{36 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{32 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{7 b^2 d x \sqrt{d+c^2 d x^2}}{1152 c^2}+\frac{43 b^2 d x^3 \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{d+c^2 d x^2}-\frac{b^2 d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{128 c^3 \sqrt{1+c^2 x^2}}-\frac{b d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{1+c^2 x^2}}-\frac{7 b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{72 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{7 b^2 d x \sqrt{d+c^2 d x^2}}{1152 c^2}+\frac{43 b^2 d x^3 \sqrt{d+c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{d+c^2 d x^2}+\frac{7 b^2 d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c^3 \sqrt{1+c^2 x^2}}-\frac{b d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c \sqrt{1+c^2 x^2}}-\frac{7 b c d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^6 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 \sqrt{1+c^2 x^2}}+\frac{d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.19683, size = 508, normalized size = 1.25 \[ \frac{-864 a^2 d^{3/2} \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+2304 a^2 c^5 d x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+4032 a^2 c^3 d x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+864 a^2 c d x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+72 b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (-12 a-3 b \sinh \left (2 \sinh ^{-1}(c x)\right )+3 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+216 a b d \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-108 a b d \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b d \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )+12 b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (-36 a \sinh \left (2 \sinh ^{-1}(c x)\right )+36 a \sinh \left (4 \sinh ^{-1}(c x)\right )+12 a \sinh \left (6 \sinh ^{-1}(c x)\right )+18 b \cosh \left (2 \sinh ^{-1}(c x)\right )-9 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )-288 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3-108 b^2 d \sqrt{c^2 d x^2+d} \sinh \left (2 \sinh ^{-1}(c x)\right )+27 b^2 d \sqrt{c^2 d x^2+d} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 d \sqrt{c^2 d x^2+d} \sinh \left (6 \sinh ^{-1}(c x)\right )}{13824 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(864*a^2*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 4032*a^2*c^3*d*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2
] + 2304*a^2*c^5*d*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 288*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 +
216*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 108*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 24*a
*b*d*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 864*a^2*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d +
c^2*d*x^2]] - 108*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 27*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSin
h[c*x]] + 4*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] + 12*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(18*b*Cos
h[2*ArcSinh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 36*a*Sinh[2*ArcSinh[c*x]] + 36*a*Sin
h[4*ArcSinh[c*x]] + 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-12*a - 3*b*Sinh[2
*ArcSinh[c*x]] + 3*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(13824*c^3*Sqrt[1 + c^2*x^2])

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Maple [B]  time = 0.358, size = 934, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*b*(d*(c^2*x^2+1))^(1/2)*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^7-1/16*a^2/c^2*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^
2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+65/3456*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*x^3+1/8*a*b*(d*(c^2*x^2+1))^(1
/2)*d/c^2/(c^2*x^2+1)*arcsinh(c*x)*x+11/12*a*b*(d*(c^2*x^2+1))^(1/2)*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^5-1/18*a
*b*(d*(c^2*x^2+1))^(1/2)*d*c^3/(c^2*x^2+1)^(1/2)*x^6-1/18*b^2*(d*(c^2*x^2+1))^(1/2)*d*c^3/(c^2*x^2+1)^(1/2)*ar
csinh(c*x)*x^6-7/48*b^2*(d*(c^2*x^2+1))^(1/2)*d*c/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^4-1/16*b^2*(d*(c^2*x^2+1))^
(1/2)*d/c/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^2-7/48*a*b*(d*(c^2*x^2+1))^(1/2)*d*c/(c^2*x^2+1)^(1/2)*x^4-1/16*a*b
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^2*d+17/24*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*ar
csinh(c*x)*x^3-1/16*a*b*(d*(c^2*x^2+1))^(1/2)*d/c/(c^2*x^2+1)^(1/2)*x^2+1/16*b^2*(d*(c^2*x^2+1))^(1/2)*d/c^2/(
c^2*x^2+1)*arcsinh(c*x)^2*x+1/6*b^2*(d*(c^2*x^2+1))^(1/2)*d*c^4/(c^2*x^2+1)*arcsinh(c*x)^2*x^7+11/24*b^2*(d*(c
^2*x^2+1))^(1/2)*d*c^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^5-1/16*a^2/c^2*d*x*(c^2*d*x^2+d)^(1/2)+1/6*a^2*x*(c^2*d*x^
2+d)^(5/2)/c^2/d-1/24*a^2/c^2*x*(c^2*d*x^2+d)^(3/2)+7/1152*b^2*(d*(c^2*x^2+1))^(1/2)*d/c^3/(c^2*x^2+1)^(1/2)*a
rcsinh(c*x)+7/1152*a*b*(d*(c^2*x^2+1))^(1/2)*d/c^3/(c^2*x^2+1)^(1/2)+1/108*b^2*(d*(c^2*x^2+1))^(1/2)*d*c^4/(c^
2*x^2+1)*x^7+59/1728*b^2*(d*(c^2*x^2+1))^(1/2)*d*c^2/(c^2*x^2+1)*x^5-7/1152*b^2*(d*(c^2*x^2+1))^(1/2)*d/c^2/(c
^2*x^2+1)*x+17/48*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)^2*x^3-1/48*b^2*(d*(c^2*x^2+1))^(1/2)/(c
^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^3*d

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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^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2,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^{2} c^{2} d x^{4} + a^{2} d x^{2} +{\left (b^{2} c^{2} d x^{4} + b^{2} d x^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{4} + a b d x^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c^2*d*x^4 + a^2*d*x^2 + (b^2*c^2*d*x^4 + b^2*d*x^2)*arcsinh(c*x)^2 + 2*(a*b*c^2*d*x^4 + a*b*d*x^
2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), 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(x**2*(c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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