\(\int x^3 (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2 \, dx\) [208]

Optimal result

Integrand size = 26, antiderivative size = 296 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {73 d^2 (a+b \text {arcsinh}(c x))^2}{3072 c^4}+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \]

[Out]

-73/3072*b^2*d^2*x^2/c^2+73/9216*b^2*d^2*x^4+43/3456*b^2*c^2*d^2*x^6+1/256*b^2*c^4*d^2*x^8-1/32*b*c*d^2*x^5*(c
^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-73/3072*d^2*(a+b*arcsinh(c*x))^2/c^4+1/24*d^2*x^4*(a+b*arcsinh(c*x))^2+1/12
*d^2*x^4*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/8*d^2*x^4*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+73/1536*b*d^2*x*(a+b*
arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-73/2304*b*d^2*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c-25/576*b*c*d^2*x^
5*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5808, 5776, 5812, 5783, 30, 5806, 14} \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {73 d^2 (a+b \text {arcsinh}(c x))^2}{3072 c^4}-\frac {1}{32} b c d^2 x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {25}{576} b c d^2 x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {73 b d^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2304 c}+\frac {73 b d^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{1536 c^3}+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {43 b^2 c^2 d^2 x^6}{3456}-\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216} \]

[In]

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

[Out]

(-73*b^2*d^2*x^2)/(3072*c^2) + (73*b^2*d^2*x^4)/9216 + (43*b^2*c^2*d^2*x^6)/3456 + (b^2*c^4*d^2*x^8)/256 + (73
*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(1536*c^3) - (73*b*d^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c
*x]))/(2304*c) - (25*b*c*d^2*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/576 - (b*c*d^2*x^5*(1 + c^2*x^2)^(3/2
)*(a + b*ArcSinh[c*x]))/32 - (73*d^2*(a + b*ArcSinh[c*x])^2)/(3072*c^4) + (d^2*x^4*(a + b*ArcSinh[c*x])^2)/24
+ (d^2*x^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/12 + (d^2*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/8

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 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*} \text {integral}& = \frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{4} \left (b c d^2\right ) \int x^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^2 \int x^3 (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{32} \left (3 b c d^2\right ) \int x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx-\frac {1}{6} \left (b c d^2\right ) \int x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{32} \left (b^2 c^2 d^2\right ) \int x^5 \left (1+c^2 x^2\right ) \, dx \\ & = -\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {1}{64} \left (b c d^2\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{36} \left (b c d^2\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{12} \left (b c d^2\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{64} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac {1}{36} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac {1}{32} \left (b^2 c^2 d^2\right ) \int \left (x^5+c^2 x^7\right ) \, dx \\ & = \frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{256} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {1}{144} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {1}{48} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {\left (3 b d^2\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{256 c}+\frac {\left (b d^2\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{48 c}+\frac {\left (b d^2\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{16 c} \\ & = \frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {\left (3 b d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{512 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{96 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{32 c^3}-\frac {\left (3 b^2 d^2\right ) \int x \, dx}{512 c^2}-\frac {\left (b^2 d^2\right ) \int x \, dx}{96 c^2}-\frac {\left (b^2 d^2\right ) \int x \, dx}{32 c^2} \\ & = -\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {73 d^2 (a+b \text {arcsinh}(c x))^2}{3072 c^4}+\frac {1}{24} d^2 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (6+8 c^2 x^2+3 c^4 x^4\right )+b^2 c x \left (-657+219 c^2 x^2+344 c^4 x^4+108 c^6 x^6\right )-6 a b \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )\right )+6 b \left (-b c x \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )+3 a \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \text {arcsinh}(c x)^2\right )}{27648 c^4} \]

[In]

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

[Out]

(d^2*(c*x*(1152*a^2*c^3*x^3*(6 + 8*c^2*x^2 + 3*c^4*x^4) + b^2*c*x*(-657 + 219*c^2*x^2 + 344*c^4*x^4 + 108*c^6*
x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x
^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 3*a*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8))*
ArcSinh[c*x] + 9*b^2*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8)*ArcSinh[c*x]^2))/(27648*c^4)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.16

method	result	size
parts	\(d^{2} a^{2} \left (\frac {1}{8} c^{4} x^{8}+\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{3}}{8}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{1536}+\frac {55 \operatorname {arcsinh}\left (c x \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )}{c^{4}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\)	\(343\)
derivativedivides	\(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{3}}{8}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{1536}+\frac {55 \operatorname {arcsinh}\left (c x \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\)	\(344\)
default	\(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{3}}{8}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{1536}+\frac {55 \operatorname {arcsinh}\left (c x \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\)	\(344\)

[In]

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

[Out]

d^2*a^2*(1/8*c^4*x^8+1/3*c^2*x^6+1/4*x^4)+d^2*b^2/c^4*(1/8*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^3-1/24*arcsinh(c
*x)^2*(c^2*x^2+1)^3-1/32*arcsinh(c*x)*c*x*(c^2*x^2+1)^(7/2)+11/576*arcsinh(c*x)*c*x*(c^2*x^2+1)^(5/2)+55/2304*
arcsinh(c*x)*c*x*(c^2*x^2+1)^(3/2)+55/1536*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+55/3072*arcsinh(c*x)^2+1/256*(c^
2*x^2+1)^4-11/3456*(c^2*x^2+1)^3-55/9216*(c^2*x^2+1)^2-55/3072*c^2*x^2-55/3072)+2*d^2*a*b/c^4*(1/8*arcsinh(c*x
)*c^8*x^8+1/3*arcsinh(c*x)*c^6*x^6+1/4*arcsinh(c*x)*c^4*x^4-1/64*c^7*x^7*(c^2*x^2+1)^(1/2)-43/1152*c^5*x^5*(c^
2*x^2+1)^(1/2)-73/4608*c^3*x^3*(c^2*x^2+1)^(1/2)+73/3072*c*x*(c^2*x^2+1)^(1/2)-73/3072*arcsinh(c*x))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.18 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {108 \, {\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{2} x^{8} + 8 \, {\left (1152 \, a^{2} + 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (2304 \, a^{2} + 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \, {\left (384 \, b^{2} c^{8} d^{2} x^{8} + 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (1152 \, a b c^{8} d^{2} x^{8} + 3072 \, a b c^{6} d^{2} x^{6} + 2304 \, a b c^{4} d^{2} x^{4} - 219 \, a b d^{2} - {\left (144 \, b^{2} c^{7} d^{2} x^{7} + 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} - 219 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (144 \, a b c^{7} d^{2} x^{7} + 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} - 219 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{27648 \, c^{4}} \]

[In]

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

[Out]

1/27648*(108*(32*a^2 + b^2)*c^8*d^2*x^8 + 8*(1152*a^2 + 43*b^2)*c^6*d^2*x^6 + 3*(2304*a^2 + 73*b^2)*c^4*d^2*x^
4 - 657*b^2*c^2*d^2*x^2 + 9*(384*b^2*c^8*d^2*x^8 + 1024*b^2*c^6*d^2*x^6 + 768*b^2*c^4*d^2*x^4 - 73*b^2*d^2)*lo
g(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(1152*a*b*c^8*d^2*x^8 + 3072*a*b*c^6*d^2*x^6 + 2304*a*b*c^4*d^2*x^4 - 219*a*b
*d^2 - (144*b^2*c^7*d^2*x^7 + 344*b^2*c^5*d^2*x^5 + 146*b^2*c^3*d^2*x^3 - 219*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*
log(c*x + sqrt(c^2*x^2 + 1)) - 6*(144*a*b*c^7*d^2*x^7 + 344*a*b*c^5*d^2*x^5 + 146*a*b*c^3*d^2*x^3 - 219*a*b*c*
d^2*x)*sqrt(c^2*x^2 + 1))/c^4

Sympy [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.74 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{8}}{8} + \frac {a^{2} c^{2} d^{2} x^{6}}{3} + \frac {a^{2} d^{2} x^{4}}{4} + \frac {a b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {a b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{32} + \frac {2 a b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 a b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{576} + \frac {a b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {73 a b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{2304 c} + \frac {73 a b d^{2} x \sqrt {c^{2} x^{2} + 1}}{1536 c^{3}} - \frac {73 a b d^{2} \operatorname {asinh}{\left (c x \right )}}{1536 c^{4}} + \frac {b^{2} c^{4} d^{2} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{4} d^{2} x^{8}}{256} - \frac {b^{2} c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{32} + \frac {b^{2} c^{2} d^{2} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {43 b^{2} c^{2} d^{2} x^{6}}{3456} - \frac {43 b^{2} c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{576} + \frac {b^{2} d^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {73 b^{2} d^{2} x^{4}}{9216} - \frac {73 b^{2} d^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2304 c} - \frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}} + \frac {73 b^{2} d^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1536 c^{3}} - \frac {73 b^{2} d^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**8/8 + a**2*c**2*d**2*x**6/3 + a**2*d**2*x**4/4 + a*b*c**4*d**2*x**8*asinh(c*x)/4
- a*b*c**3*d**2*x**7*sqrt(c**2*x**2 + 1)/32 + 2*a*b*c**2*d**2*x**6*asinh(c*x)/3 - 43*a*b*c*d**2*x**5*sqrt(c**2
*x**2 + 1)/576 + a*b*d**2*x**4*asinh(c*x)/2 - 73*a*b*d**2*x**3*sqrt(c**2*x**2 + 1)/(2304*c) + 73*a*b*d**2*x*sq
rt(c**2*x**2 + 1)/(1536*c**3) - 73*a*b*d**2*asinh(c*x)/(1536*c**4) + b**2*c**4*d**2*x**8*asinh(c*x)**2/8 + b**
2*c**4*d**2*x**8/256 - b**2*c**3*d**2*x**7*sqrt(c**2*x**2 + 1)*asinh(c*x)/32 + b**2*c**2*d**2*x**6*asinh(c*x)*
*2/3 + 43*b**2*c**2*d**2*x**6/3456 - 43*b**2*c*d**2*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/576 + b**2*d**2*x**4*a
sinh(c*x)**2/4 + 73*b**2*d**2*x**4/9216 - 73*b**2*d**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2304*c) - 73*b**2*
d**2*x**2/(3072*c**2) + 73*b**2*d**2*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(1536*c**3) - 73*b**2*d**2*asinh(c*x)**2
/(3072*c**4), Ne(c, 0)), (a**2*d**2*x**4/4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (264) = 528\).

Time = 0.25 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.57 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{8} \, b^{2} c^{4} d^{2} x^{8} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{8} \, a^{2} c^{4} d^{2} x^{8} + \frac {1}{3} \, b^{2} c^{2} d^{2} x^{6} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} c^{2} d^{2} x^{6} + \frac {1}{4} \, b^{2} d^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{1536} \, {\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^{2} + \frac {1}{9216} \, {\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^{2} + \frac {1}{4} \, a^{2} d^{2} x^{4} + \frac {1}{72} \, {\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^{2} + \frac {1}{432} \, {\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^{2} + \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^{2} + \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^{2} \]

[In]

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

[Out]

1/8*b^2*c^4*d^2*x^8*arcsinh(c*x)^2 + 1/8*a^2*c^4*d^2*x^8 + 1/3*b^2*c^2*d^2*x^6*arcsinh(c*x)^2 + 1/3*a^2*c^2*d^
2*x^6 + 1/4*b^2*d^2*x^4*arcsinh(c*x)^2 + 1/1536*(384*x^8*arcsinh(c*x) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqr
t(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^2 + 1/9216*((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^2 + 1/4*a^2*d^2*x^4 + 1/72*(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^2 + 1/432*((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^2 + 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^2 + 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*ar
csinh(c*x))*b^2*d^2

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*(c^2*d*x^2+d)^2*(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(-1)]

Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]

[In]

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

[Out]

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