3.3.17 \(\int x^3 (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\) [217]

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

[Out]

-79/5120*b^2*d^3*x^2/c^2+79/15360*b^2*d^3*x^4+401/28800*b^2*c^2*d^3*x^6+57/6400*b^2*c^4*d^3*x^8+1/500*b^2*c^6*
d^3*x^10-1/32*b*c*d^3*x^5*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-1/50*b*c*d^3*x^5*(c^2*x^2+1)^(5/2)*(a+b*arcsinh
(c*x))-79/5120*d^3*(a+b*arcsinh(c*x))^2/c^4+1/40*d^3*x^4*(a+b*arcsinh(c*x))^2+1/20*d^3*x^4*(c^2*x^2+1)*(a+b*ar
csinh(c*x))^2+3/40*d^3*x^4*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/10*d^3*x^4*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2+
79/2560*b*d^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-79/3840*b*d^3*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2
)/c-31/960*b*c*d^3*x^5*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.07, antiderivative size = 376, normalized size of antiderivative = 1.00, 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 = {5808, 5776, 5812, 5783, 30, 5806, 14, 272, 45} \begin {gather*} -\frac {79 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{5120 c^4}-\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 {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} \end {gather*}

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 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 45

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 272

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 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 5806

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[(1/(m + 2))*Simp[Sqrt[d + e*x^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/(f*(m + 2)))
*Simp[Sqrt[d + e*x^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] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 5808

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/(f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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*} \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 ) \text {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 ) \text {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 = 0.28, size = 285, normalized size = 0.76 \begin {gather*} \frac {d^3 \left (c x \left (28800 a^2 c^3 x^3 \left (10+20 c^2 x^2+15 c^4 x^4+4 c^6 x^6\right )-30 a b \sqrt {1+c^2 x^2} \left (-1185+790 c^2 x^2+3208 c^4 x^4+2736 c^6 x^6+768 c^8 x^8\right )+b^2 c x \left (-17775+5925 c^2 x^2+16040 c^4 x^4+10260 c^6 x^6+2304 c^8 x^8\right )\right )+30 b \left (-b c x \sqrt {1+c^2 x^2} \left (-1185+790 c^2 x^2+3208 c^4 x^4+2736 c^6 x^6+768 c^8 x^8\right )+15 a \left (-79+1280 c^4 x^4+2560 c^6 x^6+1920 c^8 x^8+512 c^{10} x^{10}\right )\right ) \sinh ^{-1}(c x)+225 b^2 \left (-79+1280 c^4 x^4+2560 c^6 x^6+1920 c^8 x^8+512 c^{10} x^{10}\right ) \sinh ^{-1}(c x)^2\right )}{1152000 c^4} \end {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]

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]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (336) = 672\).
time = 0.34, size = 1112, normalized size = 2.96 \begin {gather*} \frac {1}{10} \, b^{2} c^{6} d^{3} x^{10} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{10} \, a^{2} c^{6} d^{3} x^{10} + \frac {3}{8} \, b^{2} c^{4} d^{3} x^{8} \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{8} \, a^{2} c^{4} d^{3} x^{8} + \frac {1}{2} \, b^{2} c^{2} d^{3} x^{6} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} c^{2} d^{3} x^{6} + \frac {1}{6400} \, {\left (1280 \, x^{10} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{9}}{c^{2}} - \frac {144 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{6}} - \frac {210 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \operatorname {arsinh}\left (c x\right )}{c^{11}}\right )} c\right )} a b c^{6} d^{3} + \frac {1}{64000} \, {\left ({\left (\frac {128 \, x^{10}}{c^{2}} - \frac {180 \, x^{8}}{c^{4}} + \frac {280 \, x^{6}}{c^{6}} - \frac {525 \, x^{4}}{c^{8}} + \frac {1575 \, x^{2}}{c^{10}} - \frac {1575 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{12}}\right )} c^{2} - 10 \, {\left (\frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{9}}{c^{2}} - \frac {144 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{6}} - \frac {210 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \operatorname {arsinh}\left (c x\right )}{c^{11}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{6} d^{3} + \frac {1}{4} \, b^{2} d^{3} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{512} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} a b c^{4} d^{3} + \frac {1}{3072} \, {\left ({\left (\frac {36 \, x^{8}}{c^{2}} - \frac {56 \, x^{6}}{c^{4}} + \frac {105 \, x^{4}}{c^{6}} - \frac {315 \, x^{2}}{c^{8}} + \frac {315 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{10}}\right )} c^{2} - 6 \, {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{4} d^{3} + \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{48} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} a b c^{2} d^{3} + \frac {1}{288} \, {\left ({\left (\frac {8 \, x^{6}}{c^{2}} - \frac {15 \, x^{4}}{c^{4}} + \frac {45 \, x^{2}}{c^{6}} - \frac {45 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{8}}\right )} c^{2} - 6 \, {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d^{3} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b d^{3} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} d^{3} \end {gather*}

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*x)/c^11)*c)*a*b*c^6*d^3 + 1/64000*((128*x^10/c^2 - 1
80*x^8/c^4 + 280*x^6/c^6 - 525*x^4/c^8 + 1575*x^2/c^10 - 1575*log(c*x + sqrt(c^2*x^2 + 1))^2/c^12)*c^2 - 10*(1
28*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*x)/c^11)*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*x)/c^9)*c)*a*b*c^4*d^3 + 1/3
072*((36*x^8/c^2 - 56*x^6/c^4 + 105*x^4/c^6 - 315*x^2/c^8 + 315*log(c*x + 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 + 105*arcsinh(c*x)/c^9)*c*arcsinh(c*x))*b^2*c^4*d^3 + 1/4*a^2*d^3*x^4 + 1/48*(48*x^6*arcsinh(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*arcsinh(c*x)/c^
7)*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*x + 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*x)/
c^7)*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*x)/c^5)*c)*a*b*d^3 + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + 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*x)/c^5)*c*arcsinh(c*x))*b^2*d^
3

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 424, normalized size = 1.13 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 3.11, size = 654, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {a^{2} c^{6} d^{3} x^{10}}{10} + \frac {3 a^{2} c^{4} d^{3} x^{8}}{8} + \frac {a^{2} c^{2} d^{3} x^{6}}{2} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {a b c^{6} d^{3} x^{10} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {a b c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} + 1}}{50} + \frac {3 a b c^{4} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {57 a b c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{800} + a b c^{2} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )} - \frac {401 a b c d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{4800} + \frac {a b d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {79 a b d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{3840 c} + \frac {79 a b d^{3} x \sqrt {c^{2} x^{2} + 1}}{2560 c^{3}} - \frac {79 a b d^{3} \operatorname {asinh}{\left (c x \right )}}{2560 c^{4}} + \frac {b^{2} c^{6} d^{3} x^{10} \operatorname {asinh}^{2}{\left (c x \right )}}{10} + \frac {b^{2} c^{6} d^{3} x^{10}}{500} - \frac {b^{2} c^{5} d^{3} x^{9} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{50} + \frac {3 b^{2} c^{4} d^{3} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {57 b^{2} c^{4} d^{3} x^{8}}{6400} - \frac {57 b^{2} c^{3} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{800} + \frac {b^{2} c^{2} d^{3} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {401 b^{2} c^{2} d^{3} x^{6}}{28800} - \frac {401 b^{2} c d^{3} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4800} + \frac {b^{2} d^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {79 b^{2} d^{3} x^{4}}{15360} - \frac {79 b^{2} d^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3840 c} - \frac {79 b^{2} d^{3} x^{2}}{5120 c^{2}} + \frac {79 b^{2} d^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2560 c^{3}} - \frac {79 b^{2} d^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}

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))

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________