\(\int x^2 (d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2 \, dx\) [200]

Optimal result

Integrand size = 24, antiderivative size = 206 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {52 b^2 d x}{225 c^2}+\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3}-\frac {4 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c}+\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \]

[Out]

-52/225*b^2*d*x/c^2+26/675*b^2*d*x^3+2/125*b^2*c^2*d*x^5+2/15*b*d*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^3-2/2
5*b*d*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^3+2/15*d*x^3*(a+b*arcsinh(c*x))^2+1/5*d*x^3*(c^2*x^2+1)*(a+b*arcs
inh(c*x))^2+8/45*b*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-4/45*b*d*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2
)/c

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5808, 5776, 5812, 5798, 8, 30, 272, 45, 5804, 12} \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {4 b d x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{45 c}+\frac {1}{5} d x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 b d \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2 b d \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}+\frac {8 b d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{45 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {2}{125} b^2 c^2 d x^5-\frac {52 b^2 d x}{225 c^2}+\frac {26}{675} b^2 d x^3 \]

[In]

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

[Out]

(-52*b^2*d*x)/(225*c^2) + (26*b^2*d*x^3)/675 + (2*b^2*c^2*d*x^5)/125 + (8*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh
[c*x]))/(45*c^3) - (4*b*d*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(45*c) + (2*b*d*(1 + c^2*x^2)^(3/2)*(a +
b*ArcSinh[c*x]))/(15*c^3) - (2*b*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(25*c^3) + (2*d*x^3*(a + b*ArcSi
nh[c*x])^2)/15 + (d*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/5

Rule 8

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

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

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 272

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

Rule 5776

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 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5808

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^
p/(1 + c^2*x^2)^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]

Rule 5812

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} (2 d) \int x^2 (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{5} (2 b c d) \int x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = \frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{15} (4 b c d) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{5} \left (2 b^2 c^2 d\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx \\ & = -\frac {4 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c}+\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{45} \left (4 b^2 d\right ) \int x^2 \, dx+\frac {\left (2 b^2 d\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{75 c^2}+\frac {(8 b d) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{45 c} \\ & = -\frac {4 b^2 d x}{75 c^2}+\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3}-\frac {4 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c}+\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {\left (8 b^2 d\right ) \int 1 \, dx}{45 c^2} \\ & = -\frac {52 b^2 d x}{225 c^2}+\frac {26}{675} b^2 d x^3+\frac {2}{125} b^2 c^2 d x^5+\frac {8 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3}-\frac {4 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c}+\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{15 c^3}-\frac {2 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c^3}+\frac {2}{15} d x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (225 a^2 c^3 x^3 \left (5+3 c^2 x^2\right )-30 a b \sqrt {1+c^2 x^2} \left (-26+13 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (-390+65 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c^3 x^3 \left (5+3 c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (-26+13 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 c^3 x^3 \left (5+3 c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{3375 c^3} \]

[In]

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

[Out]

(d*(225*a^2*c^3*x^3*(5 + 3*c^2*x^2) - 30*a*b*Sqrt[1 + c^2*x^2]*(-26 + 13*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(-39
0 + 65*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c^3*x^3*(5 + 3*c^2*x^2) + b*Sqrt[1 + c^2*x^2]*(-26 + 13*c^2*x^2 + 9
*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c^3*x^3*(5 + 3*c^2*x^2)*ArcSinh[c*x]^2))/(3375*c^3)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.20

method	result	size
parts	\(d \,a^{2} \left (\frac {1}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 c x}{3375}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 c x \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c^{3}}+\frac {2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\)	\(247\)
derivativedivides	\(\frac {d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 c x}{3375}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 c x \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\)	\(248\)
default	\(\frac {d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 c x}{3375}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 c x \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\)	\(248\)

[In]

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

[Out]

d*a^2*(1/5*c^2*x^5+1/3*x^3)+d*b^2/c^3*(1/5*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2-2/15*arcsinh(c*x)^2*x*c-1/15*arcsi
nh(c*x)^2*c*x*(c^2*x^2+1)-2/25*arcsinh(c*x)*(c^2*x^2+1)^(5/2)-856/3375*c*x+2/125*c*x*(c^2*x^2+1)^2+22/3375*c*x
*(c^2*x^2+1)+4/15*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2/45*arcsinh(c*x)*(c^2*x^2+1)^(3/2))+2*d*a*b/c^3*(1/5*arcsinh
(c*x)*c^5*x^5+1/3*arcsinh(c*x)*c^3*x^3-1/25*c^4*x^4*(c^2*x^2+1)^(1/2)-13/225*c^2*x^2*(c^2*x^2+1)^(1/2)+26/225*
(c^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.09 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d x^{5} + 5 \, {\left (225 \, a^{2} + 26 \, b^{2}\right )} c^{3} d x^{3} - 780 \, b^{2} c d x + 225 \, {\left (3 \, b^{2} c^{5} d x^{5} + 5 \, b^{2} c^{3} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d x^{5} + 75 \, a b c^{3} d x^{3} - {\left (9 \, b^{2} c^{4} d x^{4} + 13 \, b^{2} c^{2} d x^{2} - 26 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} d x^{4} + 13 \, a b c^{2} d x^{2} - 26 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c^{3}} \]

[In]

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

[Out]

1/3375*(27*(25*a^2 + 2*b^2)*c^5*d*x^5 + 5*(225*a^2 + 26*b^2)*c^3*d*x^3 - 780*b^2*c*d*x + 225*(3*b^2*c^5*d*x^5
+ 5*b^2*c^3*d*x^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^5*d*x^5 + 75*a*b*c^3*d*x^3 - (9*b^2*c^4*d*x^4
+ 13*b^2*c^2*d*x^2 - 26*b^2*d)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*d*x^4 + 13*a*b
*c^2*d*x^2 - 26*a*b*d)*sqrt(c^2*x^2 + 1))/c^3

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.52 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{5}}{5} + \frac {a^{2} d x^{3}}{3} + \frac {2 a b c^{2} d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b c d x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {2 a b d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {26 a b d x^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {52 a b d \sqrt {c^{2} x^{2} + 1}}{225 c^{3}} + \frac {b^{2} c^{2} d x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{2} d x^{5}}{125} - \frac {2 b^{2} c d x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25} + \frac {b^{2} d x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {26 b^{2} d x^{3}}{675} - \frac {26 b^{2} d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c} - \frac {52 b^{2} d x}{225 c^{2}} + \frac {52 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*c**2*d*x**5/5 + a**2*d*x**3/3 + 2*a*b*c**2*d*x**5*asinh(c*x)/5 - 2*a*b*c*d*x**4*sqrt(c**2*x**2
+ 1)/25 + 2*a*b*d*x**3*asinh(c*x)/3 - 26*a*b*d*x**2*sqrt(c**2*x**2 + 1)/(225*c) + 52*a*b*d*sqrt(c**2*x**2 + 1
)/(225*c**3) + b**2*c**2*d*x**5*asinh(c*x)**2/5 + 2*b**2*c**2*d*x**5/125 - 2*b**2*c*d*x**4*sqrt(c**2*x**2 + 1)
*asinh(c*x)/25 + b**2*d*x**3*asinh(c*x)**2/3 + 26*b**2*d*x**3/675 - 26*b**2*d*x**2*sqrt(c**2*x**2 + 1)*asinh(c
*x)/(225*c) - 52*b**2*d*x/(225*c**2) + 52*b**2*d*sqrt(c**2*x**2 + 1)*asinh(c*x)/(225*c**3), Ne(c, 0)), (a**2*d
*x**3/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.68 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{2} d x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{2} d x^{5} + \frac {1}{3} \, b^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d + \frac {1}{3} \, a^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d \]

[In]

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

[Out]

1/5*b^2*c^2*d*x^5*arcsinh(c*x)^2 + 1/5*a^2*c^2*d*x^5 + 1/3*b^2*d*x^3*arcsinh(c*x)^2 + 2/75*(15*x^5*arcsinh(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^2*d - 2/112
5*(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*arcsinh(c*x) - (
9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d + 1/3*a^2*d*x^3 + 2/9*(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*d - 2/27*(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*d

Giac [F(-2)]

Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \]

[In]

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

[Out]

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