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

Optimal. Leaf size=214 \[ \frac{1}{5} d^2 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c}-\frac{16 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{15 c}+\frac{8}{15} d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3+\frac{298}{225} b^2 d^2 x \]

[Out]

(298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 + (2*b^2*c^4*d^2*x^5)/125 - (16*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*
ArcSinh[c*x]))/(15*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(45*c) - (2*b*d^2*(1 + c^2*x^2)^(5/
2)*(a + b*ArcSinh[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSinh[c*x])^2)/15 + (4*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[
c*x])^2)/15 + (d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5

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Rubi [A]  time = 0.255873, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5684, 5653, 5717, 8, 194} \[ \frac{1}{5} d^2 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac{8 b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c}-\frac{16 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{15 c}+\frac{8}{15} d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3+\frac{298}{225} b^2 d^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 + (2*b^2*c^4*d^2*x^5)/125 - (16*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*
ArcSinh[c*x]))/(15*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(45*c) - (2*b*d^2*(1 + c^2*x^2)^(5/
2)*(a + b*ArcSinh[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSinh[c*x])^2)/15 + (4*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[
c*x])^2)/15 + (d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5

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 5653

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{5} d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} (4 d) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} \left (2 b c d^2\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{4}{15} d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{15} \left (8 d^2\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{1}{25} \left (2 b^2 d^2\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac{1}{15} \left (8 b c d^2\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{25} \left (2 b^2 d^2\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{45} \left (8 b^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{15} \left (16 b c d^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{58}{225} b^2 d^2 x+\frac{76}{675} b^2 c^2 d^2 x^3+\frac{2}{125} b^2 c^4 d^2 x^5-\frac{16 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{15} \left (16 b^2 d^2\right ) \int 1 \, dx\\ &=\frac{298}{225} b^2 d^2 x+\frac{76}{675} b^2 c^2 d^2 x^3+\frac{2}{125} b^2 c^4 d^2 x^5-\frac{16 b d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c}-\frac{8 b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c}-\frac{2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8}{15} d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{4}{15} d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.403551, size = 191, normalized size = 0.89 \[ \frac{d^2 \left (225 a^2 c x \left (3 c^4 x^4+10 c^2 x^2+15\right )-30 a b \sqrt{c^2 x^2+1} \left (9 c^4 x^4+38 c^2 x^2+149\right )-30 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (9 c^4 x^4+38 c^2 x^2+149\right )-15 a c x \left (3 c^4 x^4+10 c^2 x^2+15\right )\right )+2 b^2 c x \left (27 c^4 x^4+190 c^2 x^2+2235\right )+225 b^2 c x \left (3 c^4 x^4+10 c^2 x^2+15\right ) \sinh ^{-1}(c x)^2\right )}{3375 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(225*a^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 2*
b^2*c*x*(2235 + 190*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 + c^2*x^2
]*(149 + 38*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]^2))/(3
375*c)

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Maple [A]  time = 0.036, size = 276, normalized size = 1.3 \begin{align*}{\frac{1}{c} \left ({d}^{2}{a}^{2} \left ({\frac{{c}^{5}{x}^{5}}{5}}+{\frac{2\,{c}^{3}{x}^{3}}{3}}+cx \right ) +{d}^{2}{b}^{2} \left ({\frac{8\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx}{15}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{5}}+{\frac{4\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) }{15}}-{\frac{298\,{\it Arcsinh} \left ( cx \right ) }{225}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{4144\,cx}{3375}}-{\frac{2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{25} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{58\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{225}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{125}}+{\frac{272\,cx \left ({c}^{2}{x}^{2}+1 \right ) }{3375}} \right ) +2\,{d}^{2}ab \left ( 1/5\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}+2/3\,{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}+{\it Arcsinh} \left ( cx \right ) cx-1/25\,{c}^{4}{x}^{4}\sqrt{{c}^{2}{x}^{2}+1}-{\frac{38\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}}{225}}-{\frac{149\,\sqrt{{c}^{2}{x}^{2}+1}}{225}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(d^2*a^2*(1/5*c^5*x^5+2/3*c^3*x^3+c*x)+d^2*b^2*(8/15*arcsinh(c*x)^2*c*x+1/5*arcsinh(c*x)^2*c*x*(c^2*x^2+1)
^2+4/15*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-298/225*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+4144/3375*c*x-2/25*arcsinh(c*x)*
c^2*x^2*(c^2*x^2+1)^(3/2)-58/225*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+2/125*c*x*(c^2*x^2+1)^2+272/3375*c*x*(
c^2*x^2+1))+2*d^2*a*b*(1/5*arcsinh(c*x)*c^5*x^5+2/3*arcsinh(c*x)*c^3*x^3+arcsinh(c*x)*c*x-1/25*c^4*x^4*(c^2*x^
2+1)^(1/2)-38/225*c^2*x^2*(c^2*x^2+1)^(1/2)-149/225*(c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.15617, size = 617, normalized size = 2.88 \begin{align*} \frac{1}{5} \, b^{2} c^{4} d^{2} x^{5} \operatorname{arsinh}\left (c x\right )^{2} + \frac{1}{5} \, a^{2} c^{4} d^{2} x^{5} + \frac{2}{3} \, b^{2} c^{2} d^{2} x^{3} \operatorname{arsinh}\left (c x\right )^{2} + \frac{2}{75} \,{\left (15 \, x^{5} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac{4 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac{2}{1125} \,{\left (15 \,{\left (\frac{3 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac{4 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname{arsinh}\left (c x\right ) - \frac{9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} + \frac{2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac{4}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} - \frac{4}{27} \,{\left (3 \, c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname{arsinh}\left (c x\right ) - \frac{c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \operatorname{arsinh}\left (c x\right )^{2} + 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*c^4*d^2*x^5*arcsinh(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 + 2/3*b^2*c^2*d^2*x^3*arcsinh(c*x)^2 + 2/75*(15*x^5*a
rcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^4
*d^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcs
inh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*c^4*d^2 + 2/3*a^2*c^2*d^2*x^3 + 4/9*(3*x^3*arcsinh(c*x) -
c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^2*d^2 - 4/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 -
2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*c^2*d^2 + b^2*d^2*x*arcsinh(c*x)^2 + 2*b^2*d
^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^2/c

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Fricas [A]  time = 2.71586, size = 616, normalized size = 2.88 \begin{align*} \frac{27 \,{\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x^{5} + 10 \,{\left (225 \, a^{2} + 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \,{\left (225 \, a^{2} + 298 \, b^{2}\right )} c d^{2} x + 225 \,{\left (3 \, b^{2} c^{5} d^{2} x^{5} + 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 30 \,{\left (45 \, a b c^{5} d^{2} x^{5} + 150 \, a b c^{3} d^{2} x^{3} + 225 \, a b c d^{2} x -{\left (9 \, b^{2} c^{4} d^{2} x^{4} + 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 30 \,{\left (9 \, a b c^{4} d^{2} x^{4} + 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{3375 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(27*(25*a^2 + 2*b^2)*c^5*d^2*x^5 + 10*(225*a^2 + 38*b^2)*c^3*d^2*x^3 + 15*(225*a^2 + 298*b^2)*c*d^2*x +
225*(3*b^2*c^5*d^2*x^5 + 10*b^2*c^3*d^2*x^3 + 15*b^2*c*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^5
*d^2*x^5 + 150*a*b*c^3*d^2*x^3 + 225*a*b*c*d^2*x - (9*b^2*c^4*d^2*x^4 + 38*b^2*c^2*d^2*x^2 + 149*b^2*d^2)*sqrt
(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*d^2*x^4 + 38*a*b*c^2*d^2*x^2 + 149*a*b*d^2)*sqrt(c
^2*x^2 + 1))/c

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Sympy [A]  time = 5.98837, size = 389, normalized size = 1.82 \begin{align*} \begin{cases} \frac{a^{2} c^{4} d^{2} x^{5}}{5} + \frac{2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac{2 a b c^{4} d^{2} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{2 a b c^{3} d^{2} x^{4} \sqrt{c^{2} x^{2} + 1}}{25} + \frac{4 a b c^{2} d^{2} x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{76 a b c d^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname{asinh}{\left (c x \right )} - \frac{298 a b d^{2} \sqrt{c^{2} x^{2} + 1}}{225 c} + \frac{b^{2} c^{4} d^{2} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}}{5} + \frac{2 b^{2} c^{4} d^{2} x^{5}}{125} - \frac{2 b^{2} c^{3} d^{2} x^{4} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{25} + \frac{2 b^{2} c^{2} d^{2} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{3} + \frac{76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac{76 b^{2} c d^{2} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname{asinh}^{2}{\left (c x \right )} + \frac{298 b^{2} d^{2} x}{225} - \frac{298 b^{2} d^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{225 c} & \text{for}\: c \neq 0 \\a^{2} d^{2} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**4*d**2*x**5/5 + 2*a**2*c**2*d**2*x**3/3 + a**2*d**2*x + 2*a*b*c**4*d**2*x**5*asinh(c*x)/5 -
2*a*b*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)/25 + 4*a*b*c**2*d**2*x**3*asinh(c*x)/3 - 76*a*b*c*d**2*x**2*sqrt(c**
2*x**2 + 1)/225 + 2*a*b*d**2*x*asinh(c*x) - 298*a*b*d**2*sqrt(c**2*x**2 + 1)/(225*c) + b**2*c**4*d**2*x**5*asi
nh(c*x)**2/5 + 2*b**2*c**4*d**2*x**5/125 - 2*b**2*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/25 + 2*b**2*c*
*2*d**2*x**3*asinh(c*x)**2/3 + 76*b**2*c**2*d**2*x**3/675 - 76*b**2*c*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)
/225 + b**2*d**2*x*asinh(c*x)**2 + 298*b**2*d**2*x/225 - 298*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(225*c),
Ne(c, 0)), (a**2*d**2*x, True))

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Giac [B]  time = 2.44213, size = 690, normalized size = 3.22 \begin{align*} \frac{1}{5} \, a^{2} c^{4} d^{2} x^{5} + \frac{2}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} a b c^{4} d^{2} + \frac{1}{1125} \,{\left (225 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{5}} - \frac{15 \,{\left (3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6}}\right )}\right )} b^{2} c^{4} d^{2} + \frac{2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac{4}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} a b c^{2} d^{2} + \frac{2}{27} \,{\left (9 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{c^{2} x^{3} - 6 \, x}{c^{3}} - \frac{3 \,{\left ({\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4}}\right )}\right )} b^{2} c^{2} d^{2} + 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} a b d^{2} +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} + 1} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} d^{2} + a^{2} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*c^4*d^2*x^5 + 2/75*(15*x^5*log(c*x + sqrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3
/2) + 15*sqrt(c^2*x^2 + 1))/c^5)*a*b*c^4*d^2 + 1/1125*(225*x^5*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((9*c^4*x^
5 - 20*c^2*x^3 + 120*x)/c^5 - 15*(3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))*log(c
*x + sqrt(c^2*x^2 + 1))/c^6))*b^2*c^4*d^2 + 2/3*a^2*c^2*d^2*x^3 + 4/9*(3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) - ((
c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*a*b*c^2*d^2 + 2/27*(9*x^3*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*
((c^2*x^3 - 6*x)/c^3 - 3*((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^4))*b^2*c^
2*d^2 + 2*(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*a*b*d^2 + (x*log(c*x + sqrt(c^2*x^2 + 1))^2 +
2*c*(x/c - sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1))/c^2))*b^2*d^2 + a^2*d^2*x