\(\int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx\) [149]

Optimal result

Integrand size = 21, antiderivative size = 195 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {3 b^2 e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d} \]

[Out]

3/4*b^4*e*(d*x+c)^2/d+3/4*b^2*e*(a+b*arcsinh(d*x+c))^2/d+3/2*b^2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2/d+1/4*e*(a
+b*arcsinh(d*x+c))^4/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^4/d-3/2*b^3*e*(d*x+c)*(a+b*arcsinh(d*x+c))*(1+(d*x
+c)^2)^(1/2)/d-b*e*(d*x+c)*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5859, 12, 5776, 5812, 5783, 30} \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}+\frac {3 b^2 e (a+b \text {arcsinh}(c+d x))^2}{4 d}-\frac {b e (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]

[In]

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

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*d) + (3*b^
2*e*(a + b*ArcSinh[c + d*x])^2)/(4*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(2*d) - (b*e*(c + d*x
)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/d + (e*(a + b*ArcSinh[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*
(a + b*ArcSinh[c + d*x])^4)/(2*d)

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 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 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 x (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d}-\frac {(2 b e) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d}-\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d} \\ & = \frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {3 b^2 e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.54 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {e \left (\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) (c+d x) \sqrt {1+(c+d x)^2}+2 a b \left (2 a^2+3 b^2\right ) \text {arcsinh}(c+d x)-2 b (c+d x) \left (-4 a^3 (c+d x)-6 a b^2 (c+d x)+6 a^2 b \sqrt {1+(c+d x)^2}+3 b^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+3 b^2 \left (2 a^2+b^2+4 a^2 (c+d x)^2+2 b^2 (c+d x)^2-4 a b (c+d x) \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+4 b^3 \left (a+2 a (c+d x)^2-b (c+d x) \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+b^4 \left (1+2 (c+d x)^2\right ) \text {arcsinh}(c+d x)^4\right )}{4 d} \]

[In]

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

[Out]

(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*(2
*a^2 + 3*b^2)*ArcSinh[c + d*x] - 2*b*(c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[1 + (c + d
*x)^2] + 3*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(2*a^2 + b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(c + d
*x)^2 - 4*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 4*b^3*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*S
qrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^4))/(4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(179)=358\).

Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.90

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(371\)
default	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(371\)
parts	\(e \,a^{4} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )}{d}+\frac {4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )}{d}+\frac {6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )}{d}+\frac {4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(381\)

[In]

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

[Out]

1/d*(1/2*e*a^4*(d*x+c)^2+e*b^4*(1/2*(1+(d*x+c)^2)*arcsinh(d*x+c)^4-(d*x+c)*arcsinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2
)-1/4*arcsinh(d*x+c)^4+3/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-3/2*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)-3/4*a
rcsinh(d*x+c)^2+3/4*(d*x+c)^2+3/4)+4*e*a*b^3*(1/2*arcsinh(d*x+c)^3*(1+(d*x+c)^2)-3/4*arcsinh(d*x+c)^2*(1+(d*x+
c)^2)^(1/2)*(d*x+c)-1/4*arcsinh(d*x+c)^3+3/4*(1+(d*x+c)^2)*arcsinh(d*x+c)-3/8*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/8*
arcsinh(d*x+c))+6*e*a^2*b^2*(1/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-1/2*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)
-1/4*arcsinh(d*x+c)^2+1/4*(d*x+c)^2+1/4)+4*e*b*a^3*(1/2*(d*x+c)^2*arcsinh(d*x+c)-1/4*(d*x+c)*(1+(d*x+c)^2)^(1/
2)+1/4*arcsinh(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (179) = 358\).

Time = 0.27 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.94 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} + b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} + a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x + {\left (2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x + {\left (2 \, a^{3} b + 3 \, a b^{3} + 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \]

[In]

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

[Out]

1/4*((2*a^4 + 6*a^2*b^2 + 3*b^4)*d^2*e*x^2 + 2*(2*a^4 + 6*a^2*b^2 + 3*b^4)*c*d*e*x + (2*b^4*d^2*e*x^2 + 4*b^4*
c*d*e*x + (2*b^4*c^2 + b^4)*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 4*(2*a*b^3*d^2*e*x^2 + 4*a
*b^3*c*d*e*x + (2*a*b^3*c^2 + a*b^3)*e - (b^4*d*e*x + b^4*c*e)*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))^3 + 3*(2*(2*a^2*b^2 + b^4)*d^2*e*x^2 + 4*(2*a^2*b^2 + b^4)*c*d*e*x + (2*a
^2*b^2 + b^4 + 2*(2*a^2*b^2 + b^4)*c^2)*e - 4*(a*b^3*d*e*x + a*b^3*c*e)*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))^2 + 2*(2*(2*a^3*b + 3*a*b^3)*d^2*e*x^2 + 4*(2*a^3*b + 3*a*b^3)*c
*d*e*x + (2*a^3*b + 3*a*b^3 + 2*(2*a^3*b + 3*a*b^3)*c^2)*e - 3*((2*a^2*b^2 + b^4)*d*e*x + (2*a^2*b^2 + b^4)*c*
e)*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)) - 2*((2*a^3*b + 3*a*b^3
)*d*e*x + (2*a^3*b + 3*a*b^3)*c*e)*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. 1027 vs. \(2 (178) = 356\).

Time = 0.45 (sec) , antiderivative size = 1027, normalized size of antiderivative = 5.27 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asinh(c + d*x)/d + 4*a**3*b*c*e*x*asinh(c + d*x) - a
**3*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + 2*a**3*b*d*e*x**2*asinh(c + d*x) - a**3*b*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1) + a**3*b*e*asinh(c + d*x)/d + 3*a**2*b**2*c**2*e*asinh(c + d*x)**2/d + 6*a**2*b**2*c*e
*x*asinh(c + d*x)**2 + 3*a**2*b**2*c*e*x - 3*a**2*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)
/d + 3*a**2*b**2*d*e*x**2*asinh(c + d*x)**2 + 3*a**2*b**2*d*e*x**2/2 - 3*a**2*b**2*e*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)*asinh(c + d*x) + 3*a**2*b**2*e*asinh(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asinh(c + d*x)**3/d + 3
*a*b**3*c**2*e*asinh(c + d*x)/d + 4*a*b**3*c*e*x*asinh(c + d*x)**3 + 6*a*b**3*c*e*x*asinh(c + d*x) - 3*a*b**3*
c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 +
1)/(2*d) + 2*a*b**3*d*e*x**2*asinh(c + d*x)**3 + 3*a*b**3*d*e*x**2*asinh(c + d*x) - 3*a*b**3*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2 - 3*a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/2 + a*b**3*e*asi
nh(c + d*x)**3/d + 3*a*b**3*e*asinh(c + d*x)/(2*d) + b**4*c**2*e*asinh(c + d*x)**4/(2*d) + 3*b**4*c**2*e*asinh
(c + d*x)**2/(2*d) + b**4*c*e*x*asinh(c + d*x)**4 + 3*b**4*c*e*x*asinh(c + d*x)**2 + 3*b**4*c*e*x/2 - b**4*c*e
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/d - 3*b**4*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*as
inh(c + d*x)/(2*d) + b**4*d*e*x**2*asinh(c + d*x)**4/2 + 3*b**4*d*e*x**2*asinh(c + d*x)**2/2 + 3*b**4*d*e*x**2
/4 - b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3 - 3*b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 + 1)*asinh(c + d*x)/2 + b**4*e*asinh(c + d*x)**4/(4*d) + 3*b**4*e*asinh(c + d*x)**2/(4*d), Ne(d, 0)), (c*e
*x*(a + b*asinh(c))**4, True))

Maxima [F]

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

[In]

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

[Out]

1/2*a^4*d*e*x^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^3*b*d*e + a^4*c*e*x + 4*((d*x + c)*arcsinh(d*x
+ c) - sqrt((d*x + c)^2 + 1))*a^3*b*c*e/d + 1/2*(b^4*d*e*x^2 + 2*b^4*c*e*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1))^4 + integrate(2*((2*(c^4*e + c^2*e)*a*b^3 + (2*a*b^3*d^4*e - b^4*d^4*e)*x^4 + 4*(2*a*b^3*c*d^3*
e - b^4*c*d^3*e)*x^3 + (2*(6*c^2*d^2*e + d^2*e)*a*b^3 - (5*c^2*d^2*e + d^2*e)*b^4)*x^2 + 2*(2*(2*c^3*d*e + c*d
*e)*a*b^3 - (c^3*d*e + c*d*e)*b^4)*x + (2*(c^3*e + c*e)*a*b^3 + (2*a*b^3*d^3*e - b^4*d^3*e)*x^3 + 3*(2*a*b^3*c
*d^2*e - b^4*c*d^2*e)*x^2 - 2*(b^4*c^2*d*e - (3*c^2*d*e + d*e)*a*b^3)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*lo
g(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + 3*(a^2*b^2*d^4*e*x^4 + 4*a^2*b^2*c*d^3*e*x^3 + (6*c^2*d^2*e
+ d^2*e)*a^2*b^2*x^2 + 2*(2*c^3*d*e + c*d*e)*a^2*b^2*x + (c^4*e + c^2*e)*a^2*b^2 + (a^2*b^2*d^3*e*x^3 + 3*a^2
*b^2*c*d^2*e*x^2 + (3*c^2*d*e + d*e)*a^2*b^2*x + (c^3*e + c*e)*a^2*b^2)*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))^2)/(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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