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

Optimal. Leaf size=376 \[ -\frac{1}{50} b c d^3 x^5 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{31}{960} b c d^3 x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{79 b d^3 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3840 c}+\frac{79 b d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2560 c^3}-\frac{79 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{5120 c^4}+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{500} b^2 c^6 d^3 x^{10}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{401 b^2 c^2 d^3 x^6}{28800}-\frac{79 b^2 d^3 x^2}{5120 c^2}+\frac{79 b^2 d^3 x^4}{15360} \]

[Out]

(-79*b^2*d^3*x^2)/(5120*c^2) + (79*b^2*d^3*x^4)/15360 + (401*b^2*c^2*d^3*x^6)/28800 + (57*b^2*c^4*d^3*x^8)/640
0 + (b^2*c^6*d^3*x^10)/500 + (79*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2560*c^3) - (79*b*d^3*x^3*Sq
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3840*c) - (31*b*c*d^3*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/960 -
(b*c*d^3*x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/32 - (b*c*d^3*x^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c
*x]))/50 - (79*d^3*(a + b*ArcSinh[c*x])^2)/(5120*c^4) + (d^3*x^4*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^
2*x^2)*(a + b*ArcSinh[c*x])^2)/20 + (3*d^3*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^2*
x^2)^3*(a + b*ArcSinh[c*x])^2)/10

________________________________________________________________________________________

Rubi [A]  time = 1.65703, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5744, 5661, 5758, 5675, 30, 5742, 14, 266, 43} \[ -\frac{1}{50} b c d^3 x^5 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{31}{960} b c d^3 x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{79 b d^3 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3840 c}+\frac{79 b d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2560 c^3}-\frac{79 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{5120 c^4}+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{500} b^2 c^6 d^3 x^{10}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{401 b^2 c^2 d^3 x^6}{28800}-\frac{79 b^2 d^3 x^2}{5120 c^2}+\frac{79 b^2 d^3 x^4}{15360} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-79*b^2*d^3*x^2)/(5120*c^2) + (79*b^2*d^3*x^4)/15360 + (401*b^2*c^2*d^3*x^6)/28800 + (57*b^2*c^4*d^3*x^8)/640
0 + (b^2*c^6*d^3*x^10)/500 + (79*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2560*c^3) - (79*b*d^3*x^3*Sq
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3840*c) - (31*b*c*d^3*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/960 -
(b*c*d^3*x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/32 - (b*c*d^3*x^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c
*x]))/50 - (79*d^3*(a + b*ArcSinh[c*x])^2)/(5120*c^4) + (d^3*x^4*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^
2*x^2)*(a + b*ArcSinh[c*x])^2)/20 + (3*d^3*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^2*
x^2)^3*(a + b*ArcSinh[c*x])^2)/10

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 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 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 30

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

Rubi steps

\begin{align*} \int x^3 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} (3 d) \int x^3 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} \left (b c d^3\right ) \int x^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} \left (3 d^2\right ) \int x^3 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{10} \left (b c d^3\right ) \int x^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{1}{20} \left (3 b c d^3\right ) \int x^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{50} \left (b^2 c^2 d^3\right ) \int x^5 \left (1+c^2 x^2\right )^2 \, dx\\ &=-\frac{1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{20} d^3 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 \int x^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{80} \left (3 b c d^3\right ) \int x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{1}{160} \left (9 b c d^3\right ) \int x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{1}{10} \left (b c d^3\right ) \int x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{100} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int x^2 \left (1+c^2 x\right )^2 \, dx,x,x^2\right )+\frac{1}{80} \left (b^2 c^2 d^3\right ) \int x^5 \left (1+c^2 x^2\right ) \, dx+\frac{1}{160} \left (3 b^2 c^2 d^3\right ) \int x^5 \left (1+c^2 x^2\right ) \, dx\\ &=-\frac{31}{960} b c d^3 x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{1}{160} \left (b c d^3\right ) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{320} \left (3 b c d^3\right ) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{60} \left (b c d^3\right ) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{20} \left (b c d^3\right ) \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{160} \left (b^2 c^2 d^3\right ) \int x^5 \, dx+\frac{1}{320} \left (3 b^2 c^2 d^3\right ) \int x^5 \, dx+\frac{1}{100} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (x^2+2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )+\frac{1}{80} \left (b^2 c^2 d^3\right ) \int \left (x^5+c^2 x^7\right ) \, dx+\frac{1}{60} \left (b^2 c^2 d^3\right ) \int x^5 \, dx+\frac{1}{160} \left (3 b^2 c^2 d^3\right ) \int \left (x^5+c^2 x^7\right ) \, dx\\ &=\frac{401 b^2 c^2 d^3 x^6}{28800}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{1}{500} b^2 c^6 d^3 x^{10}-\frac{79 b d^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3840 c}-\frac{31}{960} b c d^3 x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{640} \left (b^2 d^3\right ) \int x^3 \, dx+\frac{\left (3 b^2 d^3\right ) \int x^3 \, dx}{1280}+\frac{1}{240} \left (b^2 d^3\right ) \int x^3 \, dx+\frac{1}{80} \left (b^2 d^3\right ) \int x^3 \, dx+\frac{\left (3 b d^3\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{640 c}+\frac{\left (9 b d^3\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{1280 c}+\frac{\left (b d^3\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{80 c}+\frac{\left (3 b d^3\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{80 c}\\ &=\frac{79 b^2 d^3 x^4}{15360}+\frac{401 b^2 c^2 d^3 x^6}{28800}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{1}{500} b^2 c^6 d^3 x^{10}+\frac{79 b d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2560 c^3}-\frac{79 b d^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3840 c}-\frac{31}{960} b c d^3 x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (3 b d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{1280 c^3}-\frac{\left (9 b d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{2560 c^3}-\frac{\left (b d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{160 c^3}-\frac{\left (3 b d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{160 c^3}-\frac{\left (3 b^2 d^3\right ) \int x \, dx}{1280 c^2}-\frac{\left (9 b^2 d^3\right ) \int x \, dx}{2560 c^2}-\frac{\left (b^2 d^3\right ) \int x \, dx}{160 c^2}-\frac{\left (3 b^2 d^3\right ) \int x \, dx}{160 c^2}\\ &=-\frac{79 b^2 d^3 x^2}{5120 c^2}+\frac{79 b^2 d^3 x^4}{15360}+\frac{401 b^2 c^2 d^3 x^6}{28800}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{1}{500} b^2 c^6 d^3 x^{10}+\frac{79 b d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2560 c^3}-\frac{79 b d^3 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3840 c}-\frac{31}{960} b c d^3 x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{79 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{5120 c^4}+\frac{1}{40} d^3 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{20} d^3 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.460209, size = 285, normalized size = 0.76 \[ \frac{d^3 \left (c x \left (28800 a^2 c^3 x^3 \left (4 c^6 x^6+15 c^4 x^4+20 c^2 x^2+10\right )-30 a b \sqrt{c^2 x^2+1} \left (768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right )+b^2 c x \left (2304 c^8 x^8+10260 c^6 x^6+16040 c^4 x^4+5925 c^2 x^2-17775\right )\right )+30 b \sinh ^{-1}(c x) \left (15 a \left (512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right )-b c x \sqrt{c^2 x^2+1} \left (768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right )\right )+225 b^2 \left (512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right ) \sinh ^{-1}(c x)^2\right )}{1152000 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(c*x*(28800*a^2*c^3*x^3*(10 + 20*c^2*x^2 + 15*c^4*x^4 + 4*c^6*x^6) - 30*a*b*Sqrt[1 + c^2*x^2]*(-1185 + 79
0*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8) + b^2*c*x*(-17775 + 5925*c^2*x^2 + 16040*c^4*x^4 + 1026
0*c^6*x^6 + 2304*c^8*x^8)) + 30*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^
6 + 768*c^8*x^8)) + 15*a*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10))*ArcSinh[c*x] + 22
5*b^2*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10)*ArcSinh[c*x]^2))/(1152000*c^4)

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Maple [A]  time = 0.086, size = 508, normalized size = 1.4 \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)^3*(a+b*arcsinh(c*x))^2,x)

[Out]

1/c^4*(d^3*a^2*(1/10*c^10*x^10+3/8*c^8*x^8+1/2*c^6*x^6+1/4*c^4*x^4)+d^3*b^2*(1/10*arcsinh(c*x)^2*c^2*x^2*(c^2*
x^2+1)^4-1/40*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^3-1/40*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^2-1/40*arcsinh(c*x)
^2*c^2*x^2*(c^2*x^2+1)-1/40*arcsinh(c*x)^2*(c^2*x^2+1)-1/50*arcsinh(c*x)*c*x*(c^2*x^2+1)^(9/2)+7/800*arcsinh(c
*x)*c*x*(c^2*x^2+1)^(7/2)+49/4800*arcsinh(c*x)*c*x*(c^2*x^2+1)^(5/2)+49/3840*arcsinh(c*x)*c*x*(c^2*x^2+1)^(3/2
)+49/2560*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x+49/5120*arcsinh(c*x)^2+1/500*c^2*x^2*(c^2*x^2+1)^4+29/32000*c^2*x
^2*(c^2*x^2+1)^3-229/288000*c^2*x^2*(c^2*x^2+1)^2-4591/1152000*c^2*x^2*(c^2*x^2+1)-61/4500*c^2*x^2-61/4500)+2*
d^3*a*b*(1/10*arcsinh(c*x)*c^10*x^10+3/8*arcsinh(c*x)*c^8*x^8+1/2*arcsinh(c*x)*c^6*x^6+1/4*arcsinh(c*x)*c^4*x^
4-1/100*c^9*x^9*(c^2*x^2+1)^(1/2)-57/1600*c^7*x^7*(c^2*x^2+1)^(1/2)-401/9600*c^5*x^5*(c^2*x^2+1)^(1/2)-79/7680
*c^3*x^3*(c^2*x^2+1)^(1/2)+79/5120*c*x*(c^2*x^2+1)^(1/2)-79/5120*arcsinh(c*x)))

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Maxima [B]  time = 1.44086, size = 1669, normalized size = 4.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*c^6*d^3*x^10*arcsinh(c*x)^2 + 1/10*a^2*c^6*d^3*x^10 + 3/8*b^2*c^4*d^3*x^8*arcsinh(c*x)^2 + 3/8*a^2*c^
4*d^3*x^8 + 1/2*b^2*c^2*d^3*x^6*arcsinh(c*x)^2 + 1/2*a^2*c^2*d^3*x^6 + 1/6400*(1280*x^10*arcsinh(c*x) - (128*s
qrt(c^2*x^2 + 1)*x^9/c^2 - 144*sqrt(c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(c^2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 +
1)*x^3/c^8 + 315*sqrt(c^2*x^2 + 1)*x/c^10 - 315*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^10))*c)*a*b*c^6*d^3 + 1/
64000*((128*x^10/c^2 - 180*x^8/c^4 + 280*x^6/c^6 - 525*x^4/c^8 + 1575*x^2/c^10 - 1575*log(c^2*x/sqrt(c^2) + sq
rt(c^2*x^2 + 1))^2/c^12)*c^2 - 10*(128*sqrt(c^2*x^2 + 1)*x^9/c^2 - 144*sqrt(c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(c^
2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 + 1)*x^3/c^8 + 315*sqrt(c^2*x^2 + 1)*x/c^10 - 315*arcsinh(c^2*x/sqrt(c^2
))/(sqrt(c^2)*c^10))*c*arcsinh(c*x))*b^2*c^6*d^3 + 1/4*b^2*d^3*x^4*arcsinh(c*x)^2 + 1/512*(384*x^8*arcsinh(c*x
) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2
*x^2 + 1)*x/c^8 + 105*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c)*a*b*c^4*d^3 + 1/3072*((36*x^8/c^2 - 56*x^6/
c^4 + 105*x^4/c^6 - 315*x^2/c^8 + 315*log(c^2*x/sqrt(c^2) + sqrt(c^2*x^2 + 1))^2/c^10)*c^2 - 6*(48*sqrt(c^2*x^
2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 1
05*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c*arcsinh(c*x))*b^2*c^4*d^3 + 1/4*a^2*d^3*x^4 + 1/48*(48*x^6*arcs
inh(c*x) - (8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsi
nh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c)*a*b*c^2*d^3 + 1/288*((8*x^6/c^2 - 15*x^4/c^4 + 45*x^2/c^6 - 45*log(c^2
*x/sqrt(c^2) + sqrt(c^2*x^2 + 1))^2/c^8)*c^2 - 6*(8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 +
15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c*arcsinh(c*x))*b^2*c^2*d^3 + 1/16*
(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c^2*x/sqrt(c^2))/(s
qrt(c^2)*c^4))*c)*a*b*d^3 + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c^2*x/sqrt(c^2) + sqrt(c^2*x^2 + 1))^2/c^6)*c^2
- 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c*
arcsinh(c*x))*b^2*d^3

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Fricas [A]  time = 2.78327, size = 996, normalized size = 2.65 \begin{align*} \frac{2304 \,{\left (50 \, a^{2} + b^{2}\right )} c^{10} d^{3} x^{10} + 540 \,{\left (800 \, a^{2} + 19 \, b^{2}\right )} c^{8} d^{3} x^{8} + 40 \,{\left (14400 \, a^{2} + 401 \, b^{2}\right )} c^{6} d^{3} x^{6} + 75 \,{\left (3840 \, a^{2} + 79 \, b^{2}\right )} c^{4} d^{3} x^{4} - 17775 \, b^{2} c^{2} d^{3} x^{2} + 225 \,{\left (512 \, b^{2} c^{10} d^{3} x^{10} + 1920 \, b^{2} c^{8} d^{3} x^{8} + 2560 \, b^{2} c^{6} d^{3} x^{6} + 1280 \, b^{2} c^{4} d^{3} x^{4} - 79 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 30 \,{\left (7680 \, a b c^{10} d^{3} x^{10} + 28800 \, a b c^{8} d^{3} x^{8} + 38400 \, a b c^{6} d^{3} x^{6} + 19200 \, a b c^{4} d^{3} x^{4} - 1185 \, a b d^{3} -{\left (768 \, b^{2} c^{9} d^{3} x^{9} + 2736 \, b^{2} c^{7} d^{3} x^{7} + 3208 \, b^{2} c^{5} d^{3} x^{5} + 790 \, b^{2} c^{3} d^{3} x^{3} - 1185 \, b^{2} c d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 30 \,{\left (768 \, a b c^{9} d^{3} x^{9} + 2736 \, a b c^{7} d^{3} x^{7} + 3208 \, a b c^{5} d^{3} x^{5} + 790 \, a b c^{3} d^{3} x^{3} - 1185 \, a b c d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}}{1152000 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1152000*(2304*(50*a^2 + b^2)*c^10*d^3*x^10 + 540*(800*a^2 + 19*b^2)*c^8*d^3*x^8 + 40*(14400*a^2 + 401*b^2)*c
^6*d^3*x^6 + 75*(3840*a^2 + 79*b^2)*c^4*d^3*x^4 - 17775*b^2*c^2*d^3*x^2 + 225*(512*b^2*c^10*d^3*x^10 + 1920*b^
2*c^8*d^3*x^8 + 2560*b^2*c^6*d^3*x^6 + 1280*b^2*c^4*d^3*x^4 - 79*b^2*d^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*
(7680*a*b*c^10*d^3*x^10 + 28800*a*b*c^8*d^3*x^8 + 38400*a*b*c^6*d^3*x^6 + 19200*a*b*c^4*d^3*x^4 - 1185*a*b*d^3
- (768*b^2*c^9*d^3*x^9 + 2736*b^2*c^7*d^3*x^7 + 3208*b^2*c^5*d^3*x^5 + 790*b^2*c^3*d^3*x^3 - 1185*b^2*c*d^3*x
)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(768*a*b*c^9*d^3*x^9 + 2736*a*b*c^7*d^3*x^7 + 3208*a*b*
c^5*d^3*x^5 + 790*a*b*c^3*d^3*x^3 - 1185*a*b*c*d^3*x)*sqrt(c^2*x^2 + 1))/c^4

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Sympy [A]  time = 82.1906, size = 654, normalized size = 1.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**6*d**3*x**10/10 + 3*a**2*c**4*d**3*x**8/8 + a**2*c**2*d**3*x**6/2 + a**2*d**3*x**4/4 + a*b*
c**6*d**3*x**10*asinh(c*x)/5 - a*b*c**5*d**3*x**9*sqrt(c**2*x**2 + 1)/50 + 3*a*b*c**4*d**3*x**8*asinh(c*x)/4 -
57*a*b*c**3*d**3*x**7*sqrt(c**2*x**2 + 1)/800 + a*b*c**2*d**3*x**6*asinh(c*x) - 401*a*b*c*d**3*x**5*sqrt(c**2
*x**2 + 1)/4800 + a*b*d**3*x**4*asinh(c*x)/2 - 79*a*b*d**3*x**3*sqrt(c**2*x**2 + 1)/(3840*c) + 79*a*b*d**3*x*s
qrt(c**2*x**2 + 1)/(2560*c**3) - 79*a*b*d**3*asinh(c*x)/(2560*c**4) + b**2*c**6*d**3*x**10*asinh(c*x)**2/10 +
b**2*c**6*d**3*x**10/500 - b**2*c**5*d**3*x**9*sqrt(c**2*x**2 + 1)*asinh(c*x)/50 + 3*b**2*c**4*d**3*x**8*asinh
(c*x)**2/8 + 57*b**2*c**4*d**3*x**8/6400 - 57*b**2*c**3*d**3*x**7*sqrt(c**2*x**2 + 1)*asinh(c*x)/800 + b**2*c*
*2*d**3*x**6*asinh(c*x)**2/2 + 401*b**2*c**2*d**3*x**6/28800 - 401*b**2*c*d**3*x**5*sqrt(c**2*x**2 + 1)*asinh(
c*x)/4800 + b**2*d**3*x**4*asinh(c*x)**2/4 + 79*b**2*d**3*x**4/15360 - 79*b**2*d**3*x**3*sqrt(c**2*x**2 + 1)*a
sinh(c*x)/(3840*c) - 79*b**2*d**3*x**2/(5120*c**2) + 79*b**2*d**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2560*c**3)
- 79*b**2*d**3*asinh(c*x)**2/(5120*c**4), Ne(c, 0)), (a**2*d**3*x**4/4, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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