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

Optimal. Leaf size=204 \[ -\frac{b d^2 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}+\frac{d^2 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (c^2 x^2+1\right )^3}{108 c^2}+\frac{25}{288} b^2 d^2 x^2 \]

[Out]

(25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 + c^2*x^2)^3)/(108*c^2) - (5*b*d^2*x*Sqrt[1 + c^2
*x^2]*(a + b*ArcSinh[c*x]))/(48*c) - (5*b*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(72*c) - (b*d^2*x*(1
+ c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(18*c) - (5*d^2*(a + b*ArcSinh[c*x])^2)/(96*c^2) + (d^2*(1 + c^2*x^2)^
3*(a + b*ArcSinh[c*x])^2)/(6*c^2)

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Rubi [A]  time = 0.205952, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{b d^2 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b d^2 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}+\frac{d^2 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (c^2 x^2+1\right )^3}{108 c^2}+\frac{25}{288} b^2 d^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 + c^2*x^2)^3)/(108*c^2) - (5*b*d^2*x*Sqrt[1 + c^2
*x^2]*(a + b*ArcSinh[c*x]))/(48*c) - (5*b*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(72*c) - (b*d^2*x*(1
+ c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(18*c) - (5*d^2*(a + b*ArcSinh[c*x])^2)/(96*c^2) + (d^2*(1 + c^2*x^2)^
3*(a + b*ArcSinh[c*x])^2)/(6*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 )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}-\frac{\left (b d^2\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c}\\ &=-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{18} \left (b^2 d^2\right ) \int x \left (1+c^2 x^2\right )^2 \, dx-\frac{\left (5 b d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{18 c}\\ &=\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{72} \left (5 b^2 d^2\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac{\left (5 b d^2\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{24 c}\\ &=\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{72} \left (5 b^2 d^2\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{48} \left (5 b^2 d^2\right ) \int x \, dx-\frac{\left (5 b d^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{48 c}\\ &=\frac{25}{288} b^2 d^2 x^2+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left (1+c^2 x^2\right )^3}{108 c^2}-\frac{5 b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac{5 b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{72 c}-\frac{b d^2 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{96 c^2}+\frac{d^2 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 c^2}\\ \end{align*}

Mathematica [A]  time = 0.487445, size = 208, normalized size = 1.02 \[ \frac{d^2 \left (c x \left (144 a^2 c x \left (c^4 x^4+3 c^2 x^2+3\right )-6 a b \sqrt{c^2 x^2+1} \left (8 c^4 x^4+26 c^2 x^2+33\right )+b^2 c x \left (8 c^4 x^4+39 c^2 x^2+99\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right )-b c x \sqrt{c^2 x^2+1} \left (8 c^4 x^4+26 c^2 x^2+33\right )\right )+9 b^2 \left (16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right ) \sinh ^{-1}(c x)^2\right )}{864 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(c*x*(144*a^2*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - 6*a*b*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4) + b^2
*c*x*(99 + 39*c^2*x^2 + 8*c^4*x^4)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + 3*a*(11
+ 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6))*ArcSinh[c*x] + 9*b^2*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6)*ArcS
inh[c*x]^2))/(864*c^2)

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Maple [A]  time = 0.038, size = 324, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{2}} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{4}{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{2}}{2}} \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 ) ^{2}}{6}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{6}}-{\frac{{\it Arcsinh} \left ( cx \right ) cx}{18} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{\it Arcsinh} \left ( cx \right ) cx}{72} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{\it Arcsinh} \left ( cx \right ) cx}{48}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{5\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{96}}+{\frac{{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{108}}+{\frac{23\,{c}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }{864}}+{\frac{17\,{c}^{2}{x}^{2}}{216}}+{\frac{17}{216}} \right ) +2\,{d}^{2}ab \left ( 1/6\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}+1/2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-1/36\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-{\frac{13\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}}{144}}-{\frac{11\,cx\sqrt{{c}^{2}{x}^{2}+1}}{96}}+{\frac{11\,{\it Arcsinh} \left ( cx \right ) }{96}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(d^2*a^2*(1/6*c^6*x^6+1/2*c^4*x^4+1/2*c^2*x^2)+d^2*b^2*(1/6*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^2+1/6*arc
sinh(c*x)^2*c^2*x^2*(c^2*x^2+1)+1/6*arcsinh(c*x)^2*(c^2*x^2+1)-1/18*arcsinh(c*x)*c*x*(c^2*x^2+1)^(5/2)-5/72*ar
csinh(c*x)*c*x*(c^2*x^2+1)^(3/2)-5/48*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x-5/96*arcsinh(c*x)^2+1/108*c^2*x^2*(c^
2*x^2+1)^2+23/864*c^2*x^2*(c^2*x^2+1)+17/216*c^2*x^2+17/216)+2*d^2*a*b*(1/6*arcsinh(c*x)*c^6*x^6+1/2*arcsinh(c
*x)*c^4*x^4+1/2*arcsinh(c*x)*c^2*x^2-1/36*c^5*x^5*(c^2*x^2+1)^(1/2)-13/144*c^3*x^3*(c^2*x^2+1)^(1/2)-11/96*c*x
*(c^2*x^2+1)^(1/2)+11/96*arcsinh(c*x)))

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Maxima [B]  time = 1.23769, size = 964, normalized size = 4.73 \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)^2*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/6*b^2*c^4*d^2*x^6*arcsinh(c*x)^2 + 1/6*a^2*c^4*d^2*x^6 + 1/2*b^2*c^2*d^2*x^4*arcsinh(c*x)^2 + 1/2*a^2*c^2*d^
2*x^4 + 1/144*(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^2 + 1/864*((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*s
qrt(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^4*d^2 + 1/2*b^2*d^2*x^2*arcsinh(c*x)^2 + 1/8*(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*c^2*d^2 + 1/16*((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^2 + 1/2*a^2*d^2*x^2 + 1/
2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d^2 + 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^2

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Fricas [A]  time = 2.73979, size = 663, normalized size = 3.25 \begin{align*} \frac{8 \,{\left (18 \, a^{2} + b^{2}\right )} c^{6} d^{2} x^{6} + 3 \,{\left (144 \, a^{2} + 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \,{\left (48 \, a^{2} + 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \,{\left (16 \, b^{2} c^{6} d^{2} x^{6} + 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} + 11 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (48 \, a b c^{6} d^{2} x^{6} + 144 \, a b c^{4} d^{2} x^{4} + 144 \, a b c^{2} d^{2} x^{2} + 33 \, a b d^{2} -{\left (8 \, b^{2} c^{5} d^{2} x^{5} + 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, 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 (8 \, a b c^{5} d^{2} x^{5} + 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}}{864 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(8*(18*a^2 + b^2)*c^6*d^2*x^6 + 3*(144*a^2 + 13*b^2)*c^4*d^2*x^4 + 9*(48*a^2 + 11*b^2)*c^2*d^2*x^2 + 9*(
16*b^2*c^6*d^2*x^6 + 48*b^2*c^4*d^2*x^4 + 48*b^2*c^2*d^2*x^2 + 11*b^2*d^2)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*
(48*a*b*c^6*d^2*x^6 + 144*a*b*c^4*d^2*x^4 + 144*a*b*c^2*d^2*x^2 + 33*a*b*d^2 - (8*b^2*c^5*d^2*x^5 + 26*b^2*c^3
*d^2*x^3 + 33*b^2*c*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(8*a*b*c^5*d^2*x^5 + 26*a*b*c^3
*d^2*x^3 + 33*a*b*c*d^2*x)*sqrt(c^2*x^2 + 1))/c^2

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Sympy [A]  time = 11.8526, size = 430, normalized size = 2.11 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{6}}{6} + \frac{a^{2} c^{2} d^{2} x^{4}}{2} + \frac{a^{2} d^{2} x^{2}}{2} + \frac{a b c^{4} d^{2} x^{6} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{a b c^{3} d^{2} x^{5} \sqrt{c^{2} x^{2} + 1}}{18} + a b c^{2} d^{2} x^{4} \operatorname{asinh}{\left (c x \right )} - \frac{13 a b c d^{2} x^{3} \sqrt{c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname{asinh}{\left (c x \right )} - \frac{11 a b d^{2} x \sqrt{c^{2} x^{2} + 1}}{48 c} + \frac{11 a b d^{2} \operatorname{asinh}{\left (c x \right )}}{48 c^{2}} + \frac{b^{2} c^{4} d^{2} x^{6} \operatorname{asinh}^{2}{\left (c x \right )}}{6} + \frac{b^{2} c^{4} d^{2} x^{6}}{108} - \frac{b^{2} c^{3} d^{2} x^{5} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{18} + \frac{b^{2} c^{2} d^{2} x^{4} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{13 b^{2} c^{2} d^{2} x^{4}}{288} - \frac{13 b^{2} c d^{2} x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{72} + \frac{b^{2} d^{2} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{2} + \frac{11 b^{2} d^{2} x^{2}}{96} - \frac{11 b^{2} d^{2} x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{48 c} + \frac{11 b^{2} d^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{96 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{2} 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)**2*(a+b*asinh(c*x))**2,x)

[Out]

Piecewise((a**2*c**4*d**2*x**6/6 + a**2*c**2*d**2*x**4/2 + a**2*d**2*x**2/2 + a*b*c**4*d**2*x**6*asinh(c*x)/3
- a*b*c**3*d**2*x**5*sqrt(c**2*x**2 + 1)/18 + a*b*c**2*d**2*x**4*asinh(c*x) - 13*a*b*c*d**2*x**3*sqrt(c**2*x**
2 + 1)/72 + a*b*d**2*x**2*asinh(c*x) - 11*a*b*d**2*x*sqrt(c**2*x**2 + 1)/(48*c) + 11*a*b*d**2*asinh(c*x)/(48*c
**2) + b**2*c**4*d**2*x**6*asinh(c*x)**2/6 + b**2*c**4*d**2*x**6/108 - b**2*c**3*d**2*x**5*sqrt(c**2*x**2 + 1)
*asinh(c*x)/18 + b**2*c**2*d**2*x**4*asinh(c*x)**2/2 + 13*b**2*c**2*d**2*x**4/288 - 13*b**2*c*d**2*x**3*sqrt(c
**2*x**2 + 1)*asinh(c*x)/72 + b**2*d**2*x**2*asinh(c*x)**2/2 + 11*b**2*d**2*x**2/96 - 11*b**2*d**2*x*sqrt(c**2
*x**2 + 1)*asinh(c*x)/(48*c) + 11*b**2*d**2*asinh(c*x)**2/(96*c**2), Ne(c, 0)), (a**2*d**2*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 )}^{2}{\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)^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, x)