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

Optimal. Leaf size=265 \[ -\frac{6 c^2 \left (a^2 x^2+1\right )^{5/2}}{625 a}-\frac{272 c^2 \left (a^2 x^2+1\right )^{3/2}}{3375 a}-\frac{4144 c^2 \sqrt{a^2 x^2+1}}{1125 a}+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{1}{5} c^2 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{3 c^2 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}-\frac{4 c^2 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{5 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{298}{75} c^2 x \sinh ^{-1}(a x) \]

[Out]

(-4144*c^2*Sqrt[1 + a^2*x^2])/(1125*a) - (272*c^2*(1 + a^2*x^2)^(3/2))/(3375*a) - (6*c^2*(1 + a^2*x^2)^(5/2))/
(625*a) + (298*c^2*x*ArcSinh[a*x])/75 + (76*a^2*c^2*x^3*ArcSinh[a*x])/225 + (6*a^4*c^2*x^5*ArcSinh[a*x])/125 -
(8*c^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(5*a) - (4*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(15*a) - (3*c^2*(
1 + a^2*x^2)^(5/2)*ArcSinh[a*x]^2)/(25*a) + (8*c^2*x*ArcSinh[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcSinh[a*x]^
3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/5

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Rubi [A]  time = 0.421442, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {5684, 5653, 5717, 261, 5679, 444, 43, 194, 12, 1247, 698} \[ -\frac{6 c^2 \left (a^2 x^2+1\right )^{5/2}}{625 a}-\frac{272 c^2 \left (a^2 x^2+1\right )^{3/2}}{3375 a}-\frac{4144 c^2 \sqrt{a^2 x^2+1}}{1125 a}+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{1}{5} c^2 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{3 c^2 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}-\frac{4 c^2 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{5 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{298}{75} c^2 x \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4144*c^2*Sqrt[1 + a^2*x^2])/(1125*a) - (272*c^2*(1 + a^2*x^2)^(3/2))/(3375*a) - (6*c^2*(1 + a^2*x^2)^(5/2))/
(625*a) + (298*c^2*x*ArcSinh[a*x])/75 + (76*a^2*c^2*x^3*ArcSinh[a*x])/225 + (6*a^4*c^2*x^5*ArcSinh[a*x])/125 -
(8*c^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(5*a) - (4*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(15*a) - (3*c^2*(
1 + a^2*x^2)^(5/2)*ArcSinh[a*x]^2)/(25*a) + (8*c^2*x*ArcSinh[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcSinh[a*x]^
3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/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 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]

Rule 5679

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx-\frac{1}{5} \left (3 a c^2\right ) \int x \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2 \, dx\\ &=-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{25} \left (6 c^2\right ) \int \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x) \, dx+\frac{1}{15} \left (8 c^2\right ) \int \sinh ^{-1}(a x)^3 \, dx-\frac{1}{5} \left (4 a c^2\right ) \int x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac{6}{25} c^2 x \sinh ^{-1}(a x)+\frac{4}{25} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac{4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{15} \left (8 c^2\right ) \int \left (1+a^2 x^2\right ) \sinh ^{-1}(a x) \, dx-\frac{1}{25} \left (6 a c^2\right ) \int \frac{x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt{1+a^2 x^2}} \, dx-\frac{1}{5} \left (8 a c^2\right ) \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{58}{75} c^2 x \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac{8 c^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac{4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{5} \left (16 c^2\right ) \int \sinh ^{-1}(a x) \, dx-\frac{1}{125} \left (2 a c^2\right ) \int \frac{x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{1+a^2 x^2}} \, dx-\frac{1}{15} \left (8 a c^2\right ) \int \frac{x \left (1+\frac{a^2 x^2}{3}\right )}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{298}{75} c^2 x \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac{8 c^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac{4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3-\frac{1}{125} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{15+10 a^2 x+3 a^4 x^2}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{1}{15} \left (4 a c^2\right ) \operatorname{Subst}\left (\int \frac{1+\frac{a^2 x}{3}}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{1}{5} \left (16 a c^2\right ) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{16 c^2 \sqrt{1+a^2 x^2}}{5 a}+\frac{298}{75} c^2 x \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac{8 c^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac{4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3-\frac{1}{125} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1+a^2 x}}+4 \sqrt{1+a^2 x}+3 \left (1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )-\frac{1}{15} \left (4 a c^2\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+a^2 x}}+\frac{1}{3} \sqrt{1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{4144 c^2 \sqrt{1+a^2 x^2}}{1125 a}-\frac{272 c^2 \left (1+a^2 x^2\right )^{3/2}}{3375 a}-\frac{6 c^2 \left (1+a^2 x^2\right )^{5/2}}{625 a}+\frac{298}{75} c^2 x \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac{8 c^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac{4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.144417, size = 137, normalized size = 0.52 \[ \frac{c^2 \left (-2 \sqrt{a^2 x^2+1} \left (81 a^4 x^4+842 a^2 x^2+31841\right )+1125 a x \left (3 a^4 x^4+10 a^2 x^2+15\right ) \sinh ^{-1}(a x)^3-225 \sqrt{a^2 x^2+1} \left (9 a^4 x^4+38 a^2 x^2+149\right ) \sinh ^{-1}(a x)^2+30 a x \left (27 a^4 x^4+190 a^2 x^2+2235\right ) \sinh ^{-1}(a x)\right )}{16875 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-2*Sqrt[1 + a^2*x^2]*(31841 + 842*a^2*x^2 + 81*a^4*x^4) + 30*a*x*(2235 + 190*a^2*x^2 + 27*a^4*x^4)*ArcSi
nh[a*x] - 225*Sqrt[1 + a^2*x^2]*(149 + 38*a^2*x^2 + 9*a^4*x^4)*ArcSinh[a*x]^2 + 1125*a*x*(15 + 10*a^2*x^2 + 3*
a^4*x^4)*ArcSinh[a*x]^3))/(16875*a)

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Maple [A]  time = 0.046, size = 200, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}}{16875\,a} \left ( 3375\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{5}{x}^{5}-2025\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}{a}^{4}{x}^{4}+11250\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}+810\,{\it Arcsinh} \left ( ax \right ){a}^{5}{x}^{5}-8550\,{a}^{2}{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-162\,{a}^{4}{x}^{4}\sqrt{{a}^{2}{x}^{2}+1}+16875\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax+5700\,{\it Arcsinh} \left ( ax \right ){a}^{3}{x}^{3}-33525\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-1684\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+67050\,ax{\it Arcsinh} \left ( ax \right ) -63682\,\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16875/a*c^2*(3375*arcsinh(a*x)^3*a^5*x^5-2025*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*a^4*x^4+11250*arcsinh(a*x)^3*
a^3*x^3+810*arcsinh(a*x)*a^5*x^5-8550*a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-162*a^4*x^4*(a^2*x^2+1)^(1/2)+1
6875*arcsinh(a*x)^3*a*x+5700*arcsinh(a*x)*a^3*x^3-33525*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-1684*a^2*x^2*(a^2*x^2
+1)^(1/2)+67050*a*x*arcsinh(a*x)-63682*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.21228, size = 284, normalized size = 1.07 \begin{align*} -\frac{1}{75} \,{\left (9 \, \sqrt{a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 38 \, \sqrt{a^{2} x^{2} + 1} c^{2} x^{2} + \frac{149 \, \sqrt{a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \operatorname{arsinh}\left (a x\right )^{2} + \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname{arsinh}\left (a x\right )^{3} - \frac{2}{16875} \,{\left (81 \, \sqrt{a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 842 \, \sqrt{a^{2} x^{2} + 1} c^{2} x^{2} - \frac{15 \,{\left (27 \, a^{4} c^{2} x^{5} + 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname{arsinh}\left (a x\right )}{a} + \frac{31841 \, \sqrt{a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/75*(9*sqrt(a^2*x^2 + 1)*a^2*c^2*x^4 + 38*sqrt(a^2*x^2 + 1)*c^2*x^2 + 149*sqrt(a^2*x^2 + 1)*c^2/a^2)*a*arcsi
nh(a*x)^2 + 1/15*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arcsinh(a*x)^3 - 2/16875*(81*sqrt(a^2*x^2 + 1)*a^
2*c^2*x^4 + 842*sqrt(a^2*x^2 + 1)*c^2*x^2 - 15*(27*a^4*c^2*x^5 + 190*a^2*c^2*x^3 + 2235*c^2*x)*arcsinh(a*x)/a
+ 31841*sqrt(a^2*x^2 + 1)*c^2/a^2)*a

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Fricas [A]  time = 2.17195, size = 467, normalized size = 1.76 \begin{align*} \frac{1125 \,{\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 225 \,{\left (9 \, a^{4} c^{2} x^{4} + 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 30 \,{\left (27 \, a^{5} c^{2} x^{5} + 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 2 \,{\left (81 \, a^{4} c^{2} x^{4} + 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt{a^{2} x^{2} + 1}}{16875 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16875*(1125*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*log(a*x + sqrt(a^2*x^2 + 1))^3 - 225*(9*a^4*c^2*x^
4 + 38*a^2*c^2*x^2 + 149*c^2)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2 + 30*(27*a^5*c^2*x^5 + 190*a^3*
c^2*x^3 + 2235*a*c^2*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 2*(81*a^4*c^2*x^4 + 842*a^2*c^2*x^2 + 31841*c^2)*sqrt(a
^2*x^2 + 1))/a

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Sympy [A]  time = 9.06585, size = 262, normalized size = 0.99 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{5} \operatorname{asinh}^{3}{\left (a x \right )}}{5} + \frac{6 a^{4} c^{2} x^{5} \operatorname{asinh}{\left (a x \right )}}{125} - \frac{3 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{25} - \frac{6 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} + 1}}{625} + \frac{2 a^{2} c^{2} x^{3} \operatorname{asinh}^{3}{\left (a x \right )}}{3} + \frac{76 a^{2} c^{2} x^{3} \operatorname{asinh}{\left (a x \right )}}{225} - \frac{38 a c^{2} x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{75} - \frac{1684 a c^{2} x^{2} \sqrt{a^{2} x^{2} + 1}}{16875} + c^{2} x \operatorname{asinh}^{3}{\left (a x \right )} + \frac{298 c^{2} x \operatorname{asinh}{\left (a x \right )}}{75} - \frac{149 c^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{75 a} - \frac{63682 c^{2} \sqrt{a^{2} x^{2} + 1}}{16875 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**2*x**5*asinh(a*x)**3/5 + 6*a**4*c**2*x**5*asinh(a*x)/125 - 3*a**3*c**2*x**4*sqrt(a**2*x**2
+ 1)*asinh(a*x)**2/25 - 6*a**3*c**2*x**4*sqrt(a**2*x**2 + 1)/625 + 2*a**2*c**2*x**3*asinh(a*x)**3/3 + 76*a**2*
c**2*x**3*asinh(a*x)/225 - 38*a*c**2*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/75 - 1684*a*c**2*x**2*sqrt(a**2*x*
*2 + 1)/16875 + c**2*x*asinh(a*x)**3 + 298*c**2*x*asinh(a*x)/75 - 149*c**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(
75*a) - 63682*c**2*sqrt(a**2*x**2 + 1)/(16875*a), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.72186, size = 273, normalized size = 1.03 \begin{align*} \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + \frac{1}{16875} \,{\left (30 \,{\left (27 \, a^{4} x^{5} + 190 \, a^{2} x^{3} + 2235 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{225 \,{\left (9 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 20 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 120 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a} - \frac{2 \,{\left (81 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 680 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 31080 \, \sqrt{a^{2} x^{2} + 1}\right )}}{a}\right )} c^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 1/16875*(30*(27*a^4*x^5 + 19
0*a^2*x^3 + 2235*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 225*(9*(a^2*x^2 + 1)^(5/2) + 20*(a^2*x^2 + 1)^(3/2) + 120*s
qrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2/a - 2*(81*(a^2*x^2 + 1)^(5/2) + 680*(a^2*x^2 + 1)^(3/2) + 310
80*sqrt(a^2*x^2 + 1))/a)*c^2