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

Optimal. Leaf size=296 \[ -\frac{1}{32} b c d^2 x^5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{25}{576} b c d^2 x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{73 b d^2 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}+\frac{73 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}-\frac{73 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3072 c^4}+\frac{1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}+\frac{73 b^2 d^2 x^4}{9216} \]

[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

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Rubi [A]  time = 1.03698, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5744, 5661, 5758, 5675, 30, 5742, 14} \[ -\frac{1}{32} b c d^2 x^5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{25}{576} b c d^2 x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{12} d^2 x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{73 b d^2 x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}+\frac{73 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}-\frac{73 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3072 c^4}+\frac{1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}+\frac{73 b^2 d^2 x^4}{9216} \]

Antiderivative was successfully verified.

[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 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]]

Rubi steps

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

Mathematica [A]  time = 0.355593, size = 237, normalized size = 0.8 \[ \frac{d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (3 c^4 x^4+8 c^2 x^2+6\right )-6 a b \sqrt{c^2 x^2+1} \left (144 c^6 x^6+344 c^4 x^4+146 c^2 x^2-219\right )+b^2 c x \left (108 c^6 x^6+344 c^4 x^4+219 c^2 x^2-657\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (384 c^8 x^8+1024 c^6 x^6+768 c^4 x^4-73\right )-b c x \sqrt{c^2 x^2+1} \left (144 c^6 x^6+344 c^4 x^4+146 c^2 x^2-219\right )\right )+9 b^2 \left (384 c^8 x^8+1024 c^6 x^6+768 c^4 x^4-73\right ) \sinh ^{-1}(c x)^2\right )}{27648 c^4} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.045, size = 406, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{4}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{6}{x}^{6}}{3}}+{\frac{{c}^{4}{x}^{4}}{4}} \right ) +{d}^{2}{b}^{2} \left ({\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{8}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{24}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{24}}-{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{24}}-{\frac{{\it Arcsinh} \left ( cx \right ) cx}{32} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{\it Arcsinh} \left ( cx \right ) cx}{576} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{55\,{\it Arcsinh} \left ( cx \right ) cx}{2304} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{\it Arcsinh} \left ( cx \right ) cx}{1536}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{55\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{3072}}+{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{256}}+{\frac{5\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{6912}}-{\frac{145\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{27648}}-{\frac{5\,{c}^{2}{x}^{2}}{216}}-{\frac{5}{216}} \right ) +2\,{d}^{2}ab \left ( 1/8\,{\it Arcsinh} \left ( cx \right ){c}^{8}{x}^{8}+1/3\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}+1/4\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{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\,cx\sqrt{{c}^{2}{x}^{2}+1}}{3072}}-{\frac{73\,{\it Arcsinh} \left ( cx \right ) }{3072}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(d^2*a^2*(1/8*c^8*x^8+1/3*c^6*x^6+1/4*c^4*x^4)+d^2*b^2*(1/8*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^3-1/24*ar
csinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^2-1/24*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)-1/24*arcsinh(c*x)^2*(c^2*x^2+1)-1/3
2*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^2*x^2+1)^(1/2)*c*x+55/3072*arcsinh(c*x)^2+1/256*c^2*x^2*(c^2*x^2+1)^3+5/69
12*c^2*x^2*(c^2*x^2+1)^2-145/27648*c^2*x^2*(c^2*x^2+1)-5/216*c^2*x^2-5/216)+2*d^2*a*b*(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)))

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Maxima [B]  time = 1.33794, size = 1154, normalized size = 3.9 \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)^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^2*x/sqrt(c
^2))/(sqrt(c^2)*c^8))*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^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 + 105*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8
))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*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^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^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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*d^2 + 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^2

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Fricas [A]  time = 3.00251, size = 784, normalized size = 2.65 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 31.4014, size = 515, normalized size = 1.74 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{2}{\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)^2*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

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