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

Optimal. Leaf size=261 \[ -\frac{b d^3 x \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac{7 b d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{35 b d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac{35 b d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}+\frac{d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac{35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left (c^2 x^2+1\right )^4}{256 c^2}+\frac{7 b^2 d^3 \left (c^2 x^2+1\right )^3}{1152 c^2}+\frac{175 b^2 d^3 x^2}{3072} \]

[Out]

(175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 + c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 + c^
2*x^2)^4)/(256*c^2) - (35*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(512*c) - (35*b*d^3*x*(1 + c^2*x^2)^
(3/2)*(a + b*ArcSinh[c*x]))/(768*c) - (7*b*d^3*x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(192*c) - (b*d^3*x*
(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(32*c) - (35*d^3*(a + b*ArcSinh[c*x])^2)/(1024*c^2) + (d^3*(1 + c^2*
x^2)^4*(a + b*ArcSinh[c*x])^2)/(8*c^2)

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Rubi [A]  time = 0.256826, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5717, 5684, 5682, 5675, 30, 14, 261} \[ -\frac{b d^3 x \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac{7 b d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{35 b d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac{35 b d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}+\frac{d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac{35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left (c^2 x^2+1\right )^4}{256 c^2}+\frac{7 b^2 d^3 \left (c^2 x^2+1\right )^3}{1152 c^2}+\frac{175 b^2 d^3 x^2}{3072} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 + c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 + c^
2*x^2)^4)/(256*c^2) - (35*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(512*c) - (35*b*d^3*x*(1 + c^2*x^2)^
(3/2)*(a + b*ArcSinh[c*x]))/(768*c) - (7*b*d^3*x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(192*c) - (b*d^3*x*
(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(32*c) - (35*d^3*(a + b*ArcSinh[c*x])^2)/(1024*c^2) + (d^3*(1 + c^2*
x^2)^4*(a + b*ArcSinh[c*x])^2)/(8*c^2)

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5684

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*
(a + b*ArcSinh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(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])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1
+ c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 5682

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*
(a + b*ArcSinh[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2]), Int[(a + b*ArcSinh[c*x])^n/Sqrt[1
+ c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 + c^2*x^2]), Int[x*(a + b*ArcSinh[c*x])^(n - 1),
x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

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 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 261

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

Rubi steps

\begin{align*} \int x \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac{\left (b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 c}\\ &=-\frac{b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{32} \left (b^2 d^3\right ) \int x \left (1+c^2 x^2\right )^3 \, dx-\frac{\left (7 b d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{32 c}\\ &=\frac{b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac{7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{192} \left (7 b^2 d^3\right ) \int x \left (1+c^2 x^2\right )^2 \, dx-\frac{\left (35 b d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{192 c}\\ &=\frac{7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac{35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac{7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{768} \left (35 b^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac{\left (35 b d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{256 c}\\ &=\frac{7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac{35 b d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}-\frac{35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac{7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{768} \left (35 b^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{512} \left (35 b^2 d^3\right ) \int x \, dx-\frac{\left (35 b d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{512 c}\\ &=\frac{175 b^2 d^3 x^2}{3072}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac{b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac{35 b d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}-\frac{35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac{7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac{b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac{35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}\\ \end{align*}

Mathematica [A]  time = 0.659267, size = 256, normalized size = 0.98 \[ \frac{d^3 \left (c x \left (1152 a^2 c x \left (c^6 x^6+4 c^4 x^4+6 c^2 x^2+4\right )-6 a b \sqrt{c^2 x^2+1} \left (48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right )+b^2 c x \left (36 c^6 x^6+200 c^4 x^4+489 c^2 x^2+837\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right )-b c x \sqrt{c^2 x^2+1} \left (48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right )\right )+9 b^2 \left (128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right ) \sinh ^{-1}(c x)^2\right )}{9216 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(c*x*(1152*a^2*c*x*(4 + 6*c^2*x^2 + 4*c^4*x^4 + c^6*x^6) + b^2*c*x*(837 + 489*c^2*x^2 + 200*c^4*x^4 + 36*
c^6*x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2
*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 3*a*(93 + 512*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 12
8*c^8*x^8))*ArcSinh[c*x] + 9*b^2*(93 + 512*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8)*ArcSinh[c*x]^2))
/(9216*c^2)

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Maple [A]  time = 0.043, size = 426, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{2}} \left ({d}^{3}{a}^{2} \left ({\frac{{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{6}{x}^{6}}{2}}+{\frac{3\,{c}^{4}{x}^{4}}{4}}+{\frac{{c}^{2}{x}^{2}}{2}} \right ) +{d}^{3}{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}}{8}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{8}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{8}}-{\frac{{\it Arcsinh} \left ( cx \right ) cx}{32} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{\it Arcsinh} \left ( cx \right ) cx}{192} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{35\,{\it Arcsinh} \left ( cx \right ) cx}{768} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{\it Arcsinh} \left ( cx \right ) cx}{512}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{35\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{1024}}+{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{256}}+{\frac{23\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{2304}}+{\frac{197\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{9216}}+{\frac{{c}^{2}{x}^{2}}{18}}+{\frac{1}{18}} \right ) +2\,{d}^{3}ab \left ( 1/8\,{\it Arcsinh} \left ( cx \right ){c}^{8}{x}^{8}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}+3/4\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-{\frac{{c}^{7}{x}^{7}\sqrt{{c}^{2}{x}^{2}+1}}{64}}-{\frac{25\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}}{384}}-{\frac{163\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}}{1536}}-{\frac{93\,cx\sqrt{{c}^{2}{x}^{2}+1}}{1024}}+{\frac{93\,{\it Arcsinh} \left ( cx \right ) }{1024}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(d^3*a^2*(1/8*c^8*x^8+1/2*c^6*x^6+3/4*c^4*x^4+1/2*c^2*x^2)+d^3*b^2*(1/8*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+
1)^3+1/8*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^2+1/8*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)+1/8*arcsinh(c*x)^2*(c^2*x
^2+1)-1/32*arcsinh(c*x)*c*x*(c^2*x^2+1)^(7/2)-7/192*arcsinh(c*x)*c*x*(c^2*x^2+1)^(5/2)-35/768*arcsinh(c*x)*c*x
*(c^2*x^2+1)^(3/2)-35/512*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x-35/1024*arcsinh(c*x)^2+1/256*c^2*x^2*(c^2*x^2+1)^
3+23/2304*c^2*x^2*(c^2*x^2+1)^2+197/9216*c^2*x^2*(c^2*x^2+1)+1/18*c^2*x^2+1/18)+2*d^3*a*b*(1/8*arcsinh(c*x)*c^
8*x^8+1/2*arcsinh(c*x)*c^6*x^6+3/4*arcsinh(c*x)*c^4*x^4+1/2*arcsinh(c*x)*c^2*x^2-1/64*c^7*x^7*(c^2*x^2+1)^(1/2
)-25/384*c^5*x^5*(c^2*x^2+1)^(1/2)-163/1536*c^3*x^3*(c^2*x^2+1)^(1/2)-93/1024*c*x*(c^2*x^2+1)^(1/2)+93/1024*ar
csinh(c*x)))

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Maxima [B]  time = 1.47431, size = 1416, normalized size = 5.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*c^6*d^3*x^8*arcsinh(c*x)^2 + 1/8*a^2*c^6*d^3*x^8 + 1/2*b^2*c^4*d^3*x^6*arcsinh(c*x)^2 + 1/2*a^2*c^4*d^
3*x^6 + 3/4*b^2*c^2*d^3*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
*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/sq
rt(c^2))/(sqrt(c^2)*c^8))*c)*a*b*c^6*d^3 + 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^6*d^3 + 3/4*a^2*c^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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6
))*c)*a*b*c^4*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*a
rcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c*arcsinh(c*x))*b^2*c^4*d^3 + 1/2*b^2*d^3*x^2*arcsinh(c*x)^2 + 3/16*(
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))/(sq
rt(c^2)*c^4))*c)*a*b*c^2*d^3 + 3/32*((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*c^2*d^3 + 1/2*a^2*d^3*x^2 + 1/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsin
h(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d^3 + 1/4*(c^2*(x^2/c^2 - log(c^2*x/sqrt(c^2) + sqrt(c^2*x^2 + 1))^2/
c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2))*arcsinh(c*x))*b^2*d^3

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Fricas [A]  time = 2.82054, size = 855, normalized size = 3.28 \begin{align*} \frac{36 \,{\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{3} x^{8} + 8 \,{\left (576 \, a^{2} + 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \,{\left (2304 \, a^{2} + 163 \, b^{2}\right )} c^{4} d^{3} x^{4} + 9 \,{\left (512 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \,{\left (128 \, b^{2} c^{8} d^{3} x^{8} + 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} + 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (384 \, a b c^{8} d^{3} x^{8} + 1536 \, a b c^{6} d^{3} x^{6} + 2304 \, a b c^{4} d^{3} x^{4} + 1536 \, a b c^{2} d^{3} x^{2} + 279 \, a b d^{3} -{\left (48 \, b^{2} c^{7} d^{3} x^{7} + 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} + 279 \, 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 ) - 6 \,{\left (48 \, a b c^{7} d^{3} x^{7} + 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} + 279 \, a b c d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}}{9216 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9216*(36*(32*a^2 + b^2)*c^8*d^3*x^8 + 8*(576*a^2 + 25*b^2)*c^6*d^3*x^6 + 3*(2304*a^2 + 163*b^2)*c^4*d^3*x^4
+ 9*(512*a^2 + 93*b^2)*c^2*d^3*x^2 + 9*(128*b^2*c^8*d^3*x^8 + 512*b^2*c^6*d^3*x^6 + 768*b^2*c^4*d^3*x^4 + 512*
b^2*c^2*d^3*x^2 + 93*b^2*d^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(384*a*b*c^8*d^3*x^8 + 1536*a*b*c^6*d^3*x^6 +
2304*a*b*c^4*d^3*x^4 + 1536*a*b*c^2*d^3*x^2 + 279*a*b*d^3 - (48*b^2*c^7*d^3*x^7 + 200*b^2*c^5*d^3*x^5 + 326*b
^2*c^3*d^3*x^3 + 279*b^2*c*d^3*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(48*a*b*c^7*d^3*x^7 + 20
0*a*b*c^5*d^3*x^5 + 326*a*b*c^3*d^3*x^3 + 279*a*b*c*d^3*x)*sqrt(c^2*x^2 + 1))/c^2

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Sympy [A]  time = 35.2351, size = 573, normalized size = 2.2 \begin{align*} \begin{cases} \frac{a^{2} c^{6} d^{3} x^{8}}{8} + \frac{a^{2} c^{4} d^{3} x^{6}}{2} + \frac{3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac{a^{2} d^{3} x^{2}}{2} + \frac{a b c^{6} d^{3} x^{8} \operatorname{asinh}{\left (c x \right )}}{4} - \frac{a b c^{5} d^{3} x^{7} \sqrt{c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname{asinh}{\left (c x \right )} - \frac{25 a b c^{3} d^{3} x^{5} \sqrt{c^{2} x^{2} + 1}}{192} + \frac{3 a b c^{2} d^{3} x^{4} \operatorname{asinh}{\left (c x \right )}}{2} - \frac{163 a b c d^{3} x^{3} \sqrt{c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname{asinh}{\left (c x \right )} - \frac{93 a b d^{3} x \sqrt{c^{2} x^{2} + 1}}{512 c} + \frac{93 a b d^{3} \operatorname{asinh}{\left (c x \right )}}{512 c^{2}} + \frac{b^{2} c^{6} d^{3} x^{8} \operatorname{asinh}^{2}{\left (c x \right )}}{8} + \frac{b^{2} c^{6} d^{3} x^{8}}{256} - \frac{b^{2} c^{5} d^{3} x^{7} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{32} + \frac{b^{2} c^{4} d^{3} x^{6} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{25 b^{2} c^{4} d^{3} x^{6}}{1152} - \frac{25 b^{2} c^{3} d^{3} x^{5} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{192} + \frac{3 b^{2} c^{2} d^{3} x^{4} \operatorname{asinh}^{2}{\left (c x \right )}}{4} + \frac{163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac{163 b^{2} c d^{3} x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{768} + \frac{b^{2} d^{3} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{93 b^{2} d^{3} x^{2}}{1024} - \frac{93 b^{2} d^{3} x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{512 c} + \frac{93 b^{2} d^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{1024 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{3} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**6*d**3*x**8/8 + a**2*c**4*d**3*x**6/2 + 3*a**2*c**2*d**3*x**4/4 + a**2*d**3*x**2/2 + a*b*c*
*6*d**3*x**8*asinh(c*x)/4 - a*b*c**5*d**3*x**7*sqrt(c**2*x**2 + 1)/32 + a*b*c**4*d**3*x**6*asinh(c*x) - 25*a*b
*c**3*d**3*x**5*sqrt(c**2*x**2 + 1)/192 + 3*a*b*c**2*d**3*x**4*asinh(c*x)/2 - 163*a*b*c*d**3*x**3*sqrt(c**2*x*
*2 + 1)/768 + a*b*d**3*x**2*asinh(c*x) - 93*a*b*d**3*x*sqrt(c**2*x**2 + 1)/(512*c) + 93*a*b*d**3*asinh(c*x)/(5
12*c**2) + b**2*c**6*d**3*x**8*asinh(c*x)**2/8 + b**2*c**6*d**3*x**8/256 - b**2*c**5*d**3*x**7*sqrt(c**2*x**2
+ 1)*asinh(c*x)/32 + b**2*c**4*d**3*x**6*asinh(c*x)**2/2 + 25*b**2*c**4*d**3*x**6/1152 - 25*b**2*c**3*d**3*x**
5*sqrt(c**2*x**2 + 1)*asinh(c*x)/192 + 3*b**2*c**2*d**3*x**4*asinh(c*x)**2/4 + 163*b**2*c**2*d**3*x**4/3072 -
163*b**2*c*d**3*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/768 + b**2*d**3*x**2*asinh(c*x)**2/2 + 93*b**2*d**3*x**2/1
024 - 93*b**2*d**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(512*c) + 93*b**2*d**3*asinh(c*x)**2/(1024*c**2), Ne(c, 0)
), (a**2*d**3*x**2/2, 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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)