\(\int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx\) [139]

Optimal result

Integrand size = 23, antiderivative size = 227 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=-\frac {4}{3} a b^2 e^2 x+\frac {14 b^3 e^2 \sqrt {1+(c+d x)^2}}{9 d}-\frac {2 b^3 e^2 \left (1+(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \text {arcsinh}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d} \]

[Out]

-4/3*a*b^2*e^2*x-2/27*b^3*e^2*(1+(d*x+c)^2)^(3/2)/d-4/3*b^3*e^2*(d*x+c)*arcsinh(d*x+c)/d+2/9*b^2*e^2*(d*x+c)^3
*(a+b*arcsinh(d*x+c))/d+1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^3/d+14/9*b^3*e^2*(1+(d*x+c)^2)^(1/2)/d+2/3*b*e^
2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d-1/3*b*e^2*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/
d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5776, 5812, 5798, 5772, 267, 272, 45} \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}-\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \text {arcsinh}(c+d x)}{3 d}-\frac {2 b^3 e^2 \left ((c+d x)^2+1\right )^{3/2}}{27 d}+\frac {14 b^3 e^2 \sqrt {(c+d x)^2+1}}{9 d} \]

[In]

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

[Out]

(-4*a*b^2*e^2*x)/3 + (14*b^3*e^2*Sqrt[1 + (c + d*x)^2])/(9*d) - (2*b^3*e^2*(1 + (c + d*x)^2)^(3/2))/(27*d) - (
4*b^3*e^2*(c + d*x)*ArcSinh[c + d*x])/(3*d) + (2*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(9*d) + (2*b*e^
2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(3*d) - (b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*Ar
cSinh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^3)/(3*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 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 267

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 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 5772

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 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^2 x^2 (a+b \text {arcsinh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \text {arcsinh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arcsinh}(x))^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}+\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{3 d} \\ & = \frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}-\frac {\left (4 b^2 e^2\right ) \text {Subst}(\int (a+b \text {arcsinh}(x)) \, dx,x,c+d x)}{3 d}-\frac {\left (2 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {4}{3} a b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}-\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{9 d}-\frac {\left (4 b^3 e^2\right ) \text {Subst}(\int \text {arcsinh}(x) \, dx,x,c+d x)}{3 d} \\ & = -\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \text {arcsinh}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d}-\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,(c+d x)^2\right )}{9 d}+\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {4}{3} a b^2 e^2 x+\frac {14 b^3 e^2 \sqrt {1+(c+d x)^2}}{9 d}-\frac {2 b^3 e^2 \left (1+(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \text {arcsinh}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.14 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {e^2 \left (-12 a b^2 (c+d x)+a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {1+(c+d x)^2} \left (18 a^2+40 b^2-\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-b \left (12 b^2 (c+d x)-9 a^2 (c+d x)^3-2 b^2 (c+d x)^3-12 a b \sqrt {1+(c+d x)^2}+6 a b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)-3 b^2 \left (-3 a (c+d x)^3-2 b \sqrt {1+(c+d x)^2}+b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+3 b^3 (c+d x)^3 \text {arcsinh}(c+d x)^3\right )}{9 d} \]

[In]

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

[Out]

(e^2*(-12*a*b^2*(c + d*x) + a*(3*a^2 + 2*b^2)*(c + d*x)^3 + (b*Sqrt[1 + (c + d*x)^2]*(18*a^2 + 40*b^2 - (9*a^2
+ 2*b^2)*(c + d*x)^2))/3 - b*(12*b^2*(c + d*x) - 9*a^2*(c + d*x)^3 - 2*b^2*(c + d*x)^3 - 12*a*b*Sqrt[1 + (c +
d*x)^2] + 6*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 3*b^2*(-3*a*(c + d*x)^3 - 2*b*Sqrt[1 +
(c + d*x)^2] + b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 3*b^3*(c + d*x)^3*ArcSinh[c + d*x]^3)
)/(9*d)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(302\)
default	\(\frac {\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(302\)
parts	\(\frac {e^{2} a^{3} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )}{d}+\frac {3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {3 e^{2} a^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(310\)

[In]

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*e^2*a^3*(d*x+c)^3+e^2*b^3*(1/3*(d*x+c)^3*arcsinh(d*x+c)^3+2/3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-1/
3*(d*x+c)^2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-4/3*(d*x+c)*arcsinh(d*x+c)+40/27*(1+(d*x+c)^2)^(1/2)+2/9*(d*x
+c)^3*arcsinh(d*x+c)-2/27*(d*x+c)^2*(1+(d*x+c)^2)^(1/2))+3*e^2*a*b^2*(1/3*(d*x+c)^3*arcsinh(d*x+c)^2+4/9*arcsi
nh(d*x+c)*(1+(d*x+c)^2)^(1/2)-2/9*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^2-4/9*d*x-4/9*c+2/27*(d*x+c)^3)+3
*e^2*a^2*b*(1/3*(d*x+c)^3*arcsinh(d*x+c)-1/9*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+2/9*(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. 613 vs. \(2 (203) = 406\).

Time = 0.29 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.70 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} - 9 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \, {\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 9 \, {\left (3 \, a b^{2} d^{3} e^{2} x^{3} + 9 \, a b^{2} c d^{2} e^{2} x^{2} + 9 \, a b^{2} c^{2} d e^{2} x + 3 \, a b^{2} c^{3} e^{2} - {\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + {\left (b^{3} c^{2} - 2 \, b^{3}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{3} c - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + {\left (a b^{2} c^{2} - 2 \, a b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d e^{2} x - {\left (18 \, a^{2} b + 40 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{27 \, d} \]

[In]

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

[Out]

1/27*(3*(3*a^3 + 2*a*b^2)*d^3*e^2*x^3 + 9*(3*a^3 + 2*a*b^2)*c*d^2*e^2*x^2 - 9*(4*a*b^2 - (3*a^3 + 2*a*b^2)*c^2
)*d*e^2*x + 9*(b^3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + b^3*c^3*e^2)*log(d*x + c + sqrt(d^2
*x^2 + 2*c*d*x + c^2 + 1))^3 + 9*(3*a*b^2*d^3*e^2*x^3 + 9*a*b^2*c*d^2*e^2*x^2 + 9*a*b^2*c^2*d*e^2*x + 3*a*b^2*
c^3*e^2 - (b^3*d^2*e^2*x^2 + 2*b^3*c*d*e^2*x + (b^3*c^2 - 2*b^3)*e^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 + 3*((9*a^2*b + 2*b^3)*d^3*e^2*x^3 + 3*(9*a^2*b + 2*b^3)*c*d^2*e
^2*x^2 - 3*(4*b^3 - (9*a^2*b + 2*b^3)*c^2)*d*e^2*x - (12*b^3*c - (9*a^2*b + 2*b^3)*c^3)*e^2 - 6*(a*b^2*d^2*e^2
*x^2 + 2*a*b^2*c*d*e^2*x + (a*b^2*c^2 - 2*a*b^2)*e^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)) - ((9*a^2*b + 2*b^3)*d^2*e^2*x^2 + 2*(9*a^2*b + 2*b^3)*c*d*e^2*x - (18*a^2*b + 40*
b^3 - (9*a^2*b + 2*b^3)*c^2)*e^2)*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. 1173 vs. \(2 (211) = 422\).

Time = 0.47 (sec) , antiderivative size = 1173, normalized size of antiderivative = 5.17 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((a**3*c**2*e**2*x + a**3*c*d*e**2*x**2 + a**3*d**2*e**2*x**3/3 + a**2*b*c**3*e**2*asinh(c + d*x)/d +
3*a**2*b*c**2*e**2*x*asinh(c + d*x) - a**2*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(3*d) + 3*a**2*b*
c*d*e**2*x**2*asinh(c + d*x) - 2*a**2*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/3 + a**2*b*d**2*e**2*x**
3*asinh(c + d*x) - a**2*b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/3 + 2*a**2*b*e**2*sqrt(c**2 + 2*c*d
*x + d**2*x**2 + 1)/(3*d) + a*b**2*c**3*e**2*asinh(c + d*x)**2/d + 3*a*b**2*c**2*e**2*x*asinh(c + d*x)**2 + 2*
a*b**2*c**2*e**2*x/3 - 2*a*b**2*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + 3*a*b**2
*c*d*e**2*x**2*asinh(c + d*x)**2 + 2*a*b**2*c*d*e**2*x**2/3 - 4*a*b**2*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**
2 + 1)*asinh(c + d*x)/3 + a*b**2*d**2*e**2*x**3*asinh(c + d*x)**2 + 2*a*b**2*d**2*e**2*x**3/9 - 2*a*b**2*d*e**
2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/3 - 4*a*b**2*e**2*x/3 + 4*a*b**2*e**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + b**3*c**3*e**2*asinh(c + d*x)**3/(3*d) + 2*b**3*c**3*e**2*asinh
(c + d*x)/(9*d) + b**3*c**2*e**2*x*asinh(c + d*x)**3 + 2*b**3*c**2*e**2*x*asinh(c + d*x)/3 - b**3*c**2*e**2*sq
rt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/(3*d) - 2*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)/(27*d) + b**3*c*d*e**2*x**2*asinh(c + d*x)**3 + 2*b**3*c*d*e**2*x**2*asinh(c + d*x)/3 - 2*b**3*c*e**2*x*s
qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/3 - 4*b**3*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)
/27 - 4*b**3*c*e**2*asinh(c + d*x)/(3*d) + b**3*d**2*e**2*x**3*asinh(c + d*x)**3/3 + 2*b**3*d**2*e**2*x**3*asi
nh(c + d*x)/9 - b**3*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/3 - 2*b**3*d*e**2*x**2
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/27 - 4*b**3*e**2*x*asinh(c + d*x)/3 + 2*b**3*e**2*sqrt(c**2 + 2*c*d*x +
d**2*x**2 + 1)*asinh(c + d*x)**2/(3*d) + 40*b**3*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(27*d), Ne(d, 0)),
(c**2*e**2*x*(a + b*asinh(c))**3, True))

Maxima [F]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

1/3*a^3*d^2*e^2*x^3 + a^3*c*d*e^2*x^2 + 3/2*(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^2*b*c*d*e^2 + 1/6*(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^2*b*d^2*e^2 + a^3*c^2*e^2*x + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((
d*x + c)^2 + 1))*a^2*b*c^2*e^2/d + 1/3*(b^3*d^2*e^2*x^3 + 3*b^3*c*d*e^2*x^2 + 3*b^3*c^2*e^2*x)*log(d*x + c + s
qrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + integrate(((3*a*b^2*d^5*e^2 - b^3*d^5*e^2)*x^5 + 5*(3*a*b^2*c*d^4*e^2 -
b^3*c*d^4*e^2)*x^4 + 3*(c^5*e^2 + c^3*e^2)*a*b^2 + (3*(10*c^2*d^3*e^2 + d^3*e^2)*a*b^2 - (10*c^2*d^3*e^2 + d^3
*e^2)*b^3)*x^3 + 3*((10*c^3*d^2*e^2 + 3*c*d^2*e^2)*a*b^2 - (3*c^3*d^2*e^2 + c*d^2*e^2)*b^3)*x^2 + 3*((5*c^4*d*
e^2 + 3*c^2*d*e^2)*a*b^2 - (c^4*d*e^2 + c^2*d*e^2)*b^3)*x + ((3*a*b^2*d^4*e^2 - b^3*d^4*e^2)*x^4 + 3*(c^4*e^2
+ c^2*e^2)*a*b^2 + 4*(3*a*b^2*c*d^3*e^2 - b^3*c*d^3*e^2)*x^3 - 3*(2*b^3*c^2*d^2*e^2 - (6*c^2*d^2*e^2 + d^2*e^2
)*a*b^2)*x^2 - 3*(b^3*c^3*d*e^2 - 2*(2*c^3*d*e^2 + c*d*e^2)*a*b^2)*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))^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]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

integrate((d*e*x + c*e)^2*(b*arcsinh(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^3, x)