\(\int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx\) [127]

Optimal result

Integrand size = 23, antiderivative size = 197 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d} \]

[Out]

16/75*b^2*e^4*x-8/225*b^2*e^4*(d*x+c)^3/d+2/125*b^2*e^4*(d*x+c)^5/d+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^2/d
-16/75*b*e^4*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d+8/75*b*e^4*(d*x+c)^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2
)^(1/2)/d-2/25*b*e^4*(d*x+c)^4*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5776, 5812, 5798, 8, 30} \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d}-\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {8 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{75 d}-\frac {16 b e^4 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{75 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {16}{75} b^2 e^4 x \]

[In]

Int[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

(16*b^2*e^4*x)/75 - (8*b^2*e^4*(c + d*x)^3)/(225*d) + (2*b^2*e^4*(c + d*x)^5)/(125*d) - (16*b*e^4*Sqrt[1 + (c
+ d*x)^2]*(a + b*ArcSinh[c + d*x]))/(75*d) + (8*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x
]))/(75*d) - (2*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a
+ b*ArcSinh[c + d*x])^2)/(5*d)

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 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 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
]

Rule 5859

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^4 x^4 (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int x^4 (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d}-\frac {\left (2 b e^4\right ) \text {Subst}\left (\int \frac {x^5 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d} \\ & = -\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d}+\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (2 b^2 e^4\right ) \text {Subst}\left (\int x^4 \, dx,x,c+d x\right )}{25 d} \\ & = \frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d}-\frac {\left (16 b e^4\right ) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{75 d}-\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{75 d} \\ & = -\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d}+\frac {\left (16 b^2 e^4\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{75 d} \\ & = \frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.97 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^4 \left (240 b^2 (c+d x)-40 b^2 (c+d x)^3+9 \left (25 a^2+2 b^2\right ) (c+d x)^5+30 a b \sqrt {1+(c+d x)^2} \left (-8+4 (c+d x)^2-3 (c+d x)^4\right )+30 b \left (15 a (c+d x)^5-8 b \sqrt {1+(c+d x)^2}+4 b (c+d x)^2 \sqrt {1+(c+d x)^2}-3 b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+225 b^2 (c+d x)^5 \text {arcsinh}(c+d x)^2\right )}{1125 d} \]

[In]

Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

(e^4*(240*b^2*(c + d*x) - 40*b^2*(c + d*x)^3 + 9*(25*a^2 + 2*b^2)*(c + d*x)^5 + 30*a*b*Sqrt[1 + (c + d*x)^2]*(
-8 + 4*(c + d*x)^2 - 3*(c + d*x)^4) + 30*b*(15*a*(c + d*x)^5 - 8*b*Sqrt[1 + (c + d*x)^2] + 4*b*(c + d*x)^2*Sqr
t[1 + (c + d*x)^2] - 3*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 225*b^2*(c + d*x)^5*ArcSinh[c +
d*x]^2))/(1125*d)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\)	\(218\)
default	\(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\)	\(218\)
parts	\(\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )}{d}+\frac {2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\)	\(223\)

[In]

int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/5*e^4*a^2*(d*x+c)^5+e^4*b^2*(1/5*(d*x+c)^5*arcsinh(d*x+c)^2-16/75*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)-2/
25*(d*x+c)^4*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+8/75*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^2+16/75*d*x+16
/75*c+2/125*(d*x+c)^5-8/225*(d*x+c)^3)+2*e^4*a*b*(1/5*(d*x+c)^5*arcsinh(d*x+c)-1/25*(d*x+c)^4*(1+(d*x+c)^2)^(1
/2)+4/75*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-8/75*(1+(d*x+c)^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (177) = 354\).

Time = 0.30 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.14 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} d^{5} e^{4} x^{5} + 45 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c d^{4} e^{4} x^{4} + 10 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d^{3} e^{4} x^{3} + 30 \, {\left (3 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{3} - 4 \, b^{2} c\right )} d^{2} e^{4} x^{2} + 15 \, {\left (3 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{4} - 8 \, b^{2} c^{2} + 16 \, b^{2}\right )} d e^{4} x + 225 \, {\left (b^{2} d^{5} e^{4} x^{5} + 5 \, b^{2} c d^{4} e^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} e^{4} x^{3} + 10 \, b^{2} c^{3} d^{2} e^{4} x^{2} + 5 \, b^{2} c^{4} d e^{4} x + b^{2} c^{5} e^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 30 \, {\left (15 \, a b d^{5} e^{4} x^{5} + 75 \, a b c d^{4} e^{4} x^{4} + 150 \, a b c^{2} d^{3} e^{4} x^{3} + 150 \, a b c^{3} d^{2} e^{4} x^{2} + 75 \, a b c^{4} d e^{4} x + 15 \, a b c^{5} e^{4} - {\left (3 \, b^{2} d^{4} e^{4} x^{4} + 12 \, b^{2} c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b^{2} c^{2} - 2 \, b^{2}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b^{2} c^{3} - 2 \, b^{2} c\right )} d e^{4} x + {\left (3 \, b^{2} c^{4} - 4 \, b^{2} c^{2} + 8 \, b^{2}\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 30 \, {\left (3 \, a b d^{4} e^{4} x^{4} + 12 \, a b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, a b c^{2} - 2 \, a b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, a b c^{3} - 2 \, a b c\right )} d e^{4} x + {\left (3 \, a b c^{4} - 4 \, a b c^{2} + 8 \, a b\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{1125 \, d} \]

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")

[Out]

1/1125*(9*(25*a^2 + 2*b^2)*d^5*e^4*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*e^4*x^4 + 10*(9*(25*a^2 + 2*b^2)*c^2 - 4*b^
2)*d^3*e^4*x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*e^4*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4 - 8*b^2*c^2 +
16*b^2)*d*e^4*x + 225*(b^2*d^5*e^4*x^5 + 5*b^2*c*d^4*e^4*x^4 + 10*b^2*c^2*d^3*e^4*x^3 + 10*b^2*c^3*d^2*e^4*x^2
+ 5*b^2*c^4*d*e^4*x + b^2*c^5*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 30*(15*a*b*d^5*e^4*x^
5 + 75*a*b*c*d^4*e^4*x^4 + 150*a*b*c^2*d^3*e^4*x^3 + 150*a*b*c^3*d^2*e^4*x^2 + 75*a*b*c^4*d*e^4*x + 15*a*b*c^5
*e^4 - (3*b^2*d^4*e^4*x^4 + 12*b^2*c*d^3*e^4*x^3 + 2*(9*b^2*c^2 - 2*b^2)*d^2*e^4*x^2 + 4*(3*b^2*c^3 - 2*b^2*c)
*d*e^4*x + (3*b^2*c^4 - 4*b^2*c^2 + 8*b^2)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2
+ 2*c*d*x + c^2 + 1)) - 30*(3*a*b*d^4*e^4*x^4 + 12*a*b*c*d^3*e^4*x^3 + 2*(9*a*b*c^2 - 2*a*b)*d^2*e^4*x^2 + 4*(
3*a*b*c^3 - 2*a*b*c)*d*e^4*x + (3*a*b*c^4 - 4*a*b*c^2 + 8*a*b)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (184) = 368\).

Time = 0.63 (sec) , antiderivative size = 1268, normalized size of antiderivative = 6.44 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\text {Too large to display} \]

[In]

integrate((d*e*x+c*e)**4*(a+b*asinh(d*x+c))**2,x)

[Out]

Piecewise((a**2*c**4*e**4*x + 2*a**2*c**3*d*e**4*x**2 + 2*a**2*c**2*d**2*e**4*x**3 + a**2*c*d**3*e**4*x**4 + a
**2*d**4*e**4*x**5/5 + 2*a*b*c**5*e**4*asinh(c + d*x)/(5*d) + 2*a*b*c**4*e**4*x*asinh(c + d*x) - 2*a*b*c**4*e*
*4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 4*a*b*c**3*d*e**4*x**2*asinh(c + d*x) - 8*a*b*c**3*e**4*x*sqr
t(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a*b*c**2*d**2*e**4*x**3*asinh(c + d*x) - 12*a*b*c**2*d*e**4*x**2*sqrt
(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 8*a*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(75*d) + 2*a*b*c*d*
*3*e**4*x**4*asinh(c + d*x) - 8*a*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 16*a*b*c*e**4*x
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/75 + 2*a*b*d**4*e**4*x**5*asinh(c + d*x)/5 - 2*a*b*d**3*e**4*x**4*sqrt(c
**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 8*a*b*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/75 - 16*a*b*e**4*sq
rt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(75*d) + b**2*c**5*e**4*asinh(c + d*x)**2/(5*d) + b**2*c**4*e**4*x*asinh(c
+ d*x)**2 + 2*b**2*c**4*e**4*x/25 - 2*b**2*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(25*d
) + 2*b**2*c**3*d*e**4*x**2*asinh(c + d*x)**2 + 4*b**2*c**3*d*e**4*x**2/25 - 8*b**2*c**3*e**4*x*sqrt(c**2 + 2*
c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 + 2*b**2*c**2*d**2*e**4*x**3*asinh(c + d*x)**2 + 4*b**2*c**2*d**2*e**
4*x**3/25 - 12*b**2*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*b**2*c**2*e**4
*x/75 + 8*b**2*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(75*d) + b**2*c*d**3*e**4*x**4*as
inh(c + d*x)**2 + 2*b**2*c*d**3*e**4*x**4/25 - 8*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*as
inh(c + d*x)/25 - 8*b**2*c*d*e**4*x**2/75 + 16*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*
x)/75 + b**2*d**4*e**4*x**5*asinh(c + d*x)**2/5 + 2*b**2*d**4*e**4*x**5/125 - 2*b**2*d**3*e**4*x**4*sqrt(c**2
+ 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*b**2*d**2*e**4*x**3/225 + 8*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*
d*x + d**2*x**2 + 1)*asinh(c + d*x)/75 + 16*b**2*e**4*x/75 - 16*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)
*asinh(c + d*x)/(75*d), Ne(d, 0)), (c**4*e**4*x*(a + b*asinh(c))**2, True))

Maxima [F]

\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")

[Out]

1/5*a^2*d^4*e^4*x^5 + a^2*c*d^3*e^4*x^4 + 2*a^2*c^2*d^2*e^4*x^3 + 2*a^2*c^3*d*e^4*x^2 + 2*(2*x^2*arcsinh(d*x +
c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*
d*x + c^2 + 1)*c/d^3))*a*b*c^3*d*e^4 + 2/3*(6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^
2/d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^
2 + 1)*c*x/d^3 + 9*(c^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x
^2 + 2*c*d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a*b*c^2*d^2*e^4 + 1/12*(
24*x^4*arcsinh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*
c*x^2/d^3 + 105*c^4*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*
x + c^2 + 1)*c^2*x/d^4 - 90*(c^2 + 1)*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 10
5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x/d^4 + 9*(c^2 + 1
)^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^
2 + 1)*c/d^5)*d)*a*b*c*d^3*e^4 + 1/300*(120*x^5*arcsinh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^4/d
^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^3/d^3 + 126*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x^2/d^4 - 945*
c^5*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 - 315*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^
3*x/d^5 - 32*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x^2/d^4 + 1050*(c^2 + 1)*c^3*arcsinh(2*(d^2*x + c*d)/
sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 + 945*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^4/d^6 + 161*sqrt(d^2*x^2 + 2
*c*d*x + c^2 + 1)*(c^2 + 1)*c*x/d^5 - 225*(c^2 + 1)^2*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*
d^2))/d^6 - 735*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c^2/d^6 + 64*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^
2 + 1)^2/d^6)*d)*a*b*d^4*e^4 + a^2*c^4*e^4*x + 2*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a*b*c^4*
e^4/d + 1/5*(b^2*d^4*e^4*x^5 + 5*b^2*c*d^3*e^4*x^4 + 10*b^2*c^2*d^2*e^4*x^3 + 10*b^2*c^3*d*e^4*x^2 + 5*b^2*c^4
*e^4*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - integrate(2/5*(b^2*d^7*e^4*x^7 + 7*b^2*c*d^6*e^4*
x^6 + (21*c^2*d^5*e^4 + d^5*e^4)*b^2*x^5 + 5*(7*c^3*d^4*e^4 + c*d^4*e^4)*b^2*x^4 + 5*(7*c^4*d^3*e^4 + 2*c^2*d^
3*e^4)*b^2*x^3 + 10*(2*c^5*d^2*e^4 + c^3*d^2*e^4)*b^2*x^2 + 5*(c^6*d*e^4 + c^4*d*e^4)*b^2*x + (b^2*d^6*e^4*x^6
+ 6*b^2*c*d^5*e^4*x^5 + 15*b^2*c^2*d^4*e^4*x^4 + 20*b^2*c^3*d^3*e^4*x^3 + 15*b^2*c^4*d^2*e^4*x^2 + 5*b^2*c^5*
d*e^4*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^3*x^3 + 3*c*d^
2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x)

Giac [F]

\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^2, x)