\(\int x^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [137]

Optimal result

Integrand size = 26, antiderivative size = 337 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {5 b d^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}+\frac {5 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

5/48*d*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/8*x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))+5/128*d^2*x*(
a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)-5/256*b*d^2*x^2*
(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-59/768*b*c*d^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-17/288*b*c^3*
d^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*x^8*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/256*d
^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5808, 5806, 5812, 5783, 30, 14, 272, 45} \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {5 d^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {5 d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {c^2 x^2+1}}-\frac {5 b d^2 x^2 \sqrt {c^2 d x^2+d}}{256 c \sqrt {c^2 x^2+1}}-\frac {59 b c d^2 x^4 \sqrt {c^2 d x^2+d}}{768 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^8 \sqrt {c^2 d x^2+d}}{64 \sqrt {c^2 x^2+1}}-\frac {17 b c^3 d^2 x^6 \sqrt {c^2 d x^2+d}}{288 \sqrt {c^2 x^2+1}} \]

[In]

Int[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-5*b*d^2*x^2*Sqrt[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d + c^2*d*x^2])/(768*Sqrt[
1 + c^2*x^2]) - (17*b*c^3*d^2*x^6*Sqrt[d + c^2*d*x^2])/(288*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d + c^2*d
*x^2])/(64*Sqrt[1 + c^2*x^2]) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (5*d^2*x^3*Sqrt
[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/64 + (5*d*x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/48 + (x^3*(d +
c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/8 - (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt
[1 + c^2*x^2])

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 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 5783

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

Rule 5806

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

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}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} (5 d) \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{16} \left (5 d^2\right ) \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{48 \sqrt {1+c^2 x^2}} \\ & = \frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (x+2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{48 \sqrt {1+c^2 x^2}} \\ & = -\frac {59 b c d^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}+\frac {5 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b d^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1+c^2 x^2}} \\ & = -\frac {5 b d^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}+\frac {5 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.15 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (2880 a c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+22656 a c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+26112 a c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+9216 a c^7 x^7 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-1440 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2+576 b \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-144 b \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-64 b \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))-9 b \sqrt {d+c^2 d x^2} \cosh (8 \text {arcsinh}(c x))-2880 a \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+24 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (-48 \sinh (2 \text {arcsinh}(c x))+24 \sinh (4 \text {arcsinh}(c x))+16 \sinh (6 \text {arcsinh}(c x))+3 \sinh (8 \text {arcsinh}(c x)))\right )}{73728 c^3 \sqrt {1+c^2 x^2}} \]

[In]

Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d^2*(2880*a*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 22656*a*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]
+ 26112*a*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 9216*a*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2
] - 1440*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 + 576*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 144*b*Sqrt[d
+ c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 64*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 9*b*Sqrt[d + c^2*d*x^2]*Co
sh[8*ArcSinh[c*x]] - 2880*a*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 24*b*Sqrt[d +
c^2*d*x^2]*ArcSinh[c*x]*(-48*Sinh[2*ArcSinh[c*x]] + 24*Sinh[4*ArcSinh[c*x]] + 16*Sinh[6*ArcSinh[c*x]] + 3*Sin
h[8*ArcSinh[c*x]])))/(73728*c^3*Sqrt[1 + c^2*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1164\) vs. \(2(291)=582\).

Time = 0.22 (sec) , antiderivative size = 1165, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1165\)
parts	\(\text {Expression too large to display}\)	\(1165\)

[In]

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/8*a*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/48*a/c^2*x*(c^2*d*x^2+d)^(5/2)-5/192*a/c^2*d*x*(c^2*d*x^2+d)^(3/2)-5/128*a
/c^2*d^2*x*(c^2*d*x^2+d)^(1/2)-5/128*a/c^2*d^3*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(
-5/256*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^2*d^2+1/16384*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x
^9+128*c^8*x^8*(c^2*x^2+1)^(1/2)+320*c^7*x^7+256*c^6*x^6*(c^2*x^2+1)^(1/2)+272*c^5*x^5+160*c^4*x^4*(c^2*x^2+1)
^(1/2)+88*c^3*x^3+32*c^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x+(c^2*x^2+1)^(1/2))*(-1+8*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1
)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2
)+38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(-1+6*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/1
024*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+
(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c
^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(
2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/1024*(
d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*
x^2+1)^(1/2))*(1+4*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*c^6*x^6*(c^2*
x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)
^(1/2))*(1+6*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/16384*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x^9-128*c^8*x^8*(c^2*x^2
+1)^(1/2)+320*c^7*x^7-256*c^6*x^6*(c^2*x^2+1)^(1/2)+272*c^5*x^5-160*c^4*x^4*(c^2*x^2+1)^(1/2)+88*c^3*x^3-32*c^
2*x^2*(c^2*x^2+1)^(1/2)+8*c*x-(c^2*x^2+1)^(1/2))*(1+8*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1))

Fricas [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^6 + 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 + 2*b*c^2*d^2*x^4 + b*d^2*x^2)*arcsinh(
c*x))*sqrt(c^2*d*x^2 + d), x)

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]

[In]

int(x^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)

[Out]

int(x^2*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)