\(\int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^2} \, dx\) [163]

Optimal result

Integrand size = 23, antiderivative size = 188 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5859, 12, 5778, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 (c+d x)^3 \sqrt {(c+d x)^2+1}}{b d (a+b \text {arcsinh}(c+d x))} \]

[In]

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

[Out]

-((e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) - (e^3*Cosh[(2*a)/b]*CoshIntegral[(2
*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d) + (e^3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(2
*b^2*d) + (e^3*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d) - (e^3*Sinh[(4*a)/b]*Sinh
Integral[(4*(a + b*ArcSinh[c + d*x]))/b])/(2*b^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 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5778

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.03 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^3 \left (\frac {2 b (c+d x)^3 \sqrt {1+(c+d x)^2}}{a+b \text {arcsinh}(c+d x)}+\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-3 \log (a+b \text {arcsinh}(c+d x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+3 \left (\log (a+b \text {arcsinh}(c+d x))+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )+\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )}{2 b^2 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs. \(2(178)=356\).

Time = 0.91 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.06

method	result	size
derivativedivides	\(\frac {\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3}}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{4 b^{2}}-\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{4 b^{2}}}{d}\)	\(388\)
default	\(\frac {\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3}}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{4 b^{2}}-\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{4 b^{2}}}{d}\)	\(388\)

[In]

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

[Out]

1/d*(1/16*(-8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+8*(d*x+c)^4-4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+8*(d*x+c)^2+1)*e^3/b/(a+
b*arcsinh(d*x+c))-1/4*e^3/b^2*exp(4*a/b)*Ei(1,4*arcsinh(d*x+c)+4*a/b)-1/8*(-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*(d
*x+c)^2+1)*e^3/b/(a+b*arcsinh(d*x+c))+1/4*e^3/b^2*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)+1/8/b*e^3*(2*(d*x+c)
^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/4/b^2*e^3*exp(-2*a/b)*Ei(1,-2*arcsinh(d*x+c)-2*a/b)
-1/16/b*e^3*(8*(d*x+c)^4+8*(d*x+c)^2+8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1)/(a+b*arc
sinh(d*x+c))-1/4/b^2*e^3*exp(-4*a/b)*Ei(1,-4*arcsinh(d*x+c)-4*a/b))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 + c^4*e^3 + (15*c^2*d^4*e^3 + d^4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 + c*d^
3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*x^2 + 2*(3*c^5*d*e^3 + 2*c^3*d*e^3)*x + (d^5*e^3*x^5 + 5*c*d^4*
e^3*x^4 + c^5*e^3 + c^3*e^3 + (10*c^2*d^3*e^3 + d^3*e^3)*x^3 + (10*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + (5*c^4*d*e
^3 + 3*c^2*d*e^3)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d + d)*a*b + (b^2*
d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d + d)*b^2 + (b^2*d^2*x + b^2*c*d)*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)) + (a*b*d^2*x + a*b*c*d)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate
((4*d^7*e^3*x^7 + 28*c*d^6*e^3*x^6 + 4*c^7*e^3 + 8*c^5*e^3 + 4*c^3*e^3 + 4*(21*c^2*d^5*e^3 + 2*d^5*e^3)*x^5 +
20*(7*c^3*d^4*e^3 + 2*c*d^4*e^3)*x^4 + 4*(35*c^4*d^3*e^3 + 20*c^2*d^3*e^3 + d^3*e^3)*x^3 + 4*(21*c^5*d^2*e^3 +
20*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + 2*(2*d^5*e^3*x^5 + 10*c*d^4*e^3*x^4 + 2*c^5*e^3 + c^3*e^3 + (20*c^2*d^3*e
^3 + d^3*e^3)*x^3 + (20*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + (10*c^4*d*e^3 + 3*c^2*d*e^3)*x)*(d^2*x^2 + 2*c*d*x +
c^2 + 1) + 4*(7*c^6*d*e^3 + 10*c^4*d*e^3 + 3*c^2*d*e^3)*x + (8*d^6*e^3*x^6 + 48*c*d^5*e^3*x^5 + 8*c^6*e^3 + 10
*c^4*e^3 + 3*c^2*e^3 + 10*(12*c^2*d^4*e^3 + d^4*e^3)*x^4 + 40*(4*c^3*d^3*e^3 + c*d^3*e^3)*x^3 + 3*(40*c^4*d^2*
e^3 + 20*c^2*d^2*e^3 + d^2*e^3)*x^2 + 2*(24*c^5*d*e^3 + 20*c^3*d*e^3 + 3*c*d*e^3)*x)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1))/(a*b*d^4*x^4 + 4*a*b*c*d^3*x^3 + 2*(3*c^2*d^2 + d^2)*a*b*x^2 + 4*(c^3*d + c*d)*a*b*x + (c^4 + 2*c^2
+ 1)*a*b + (a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^
3 + 2*(3*c^2*d^2 + d^2)*b^2*x^2 + 4*(c^3*d + c*d)*b^2*x + (c^4 + 2*c^2 + 1)*b^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d + d)*b^2*x + (c^3 + c)*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*(a*b*d^3*x^3 + 3*a
*b*c*d^2*x^2 + (3*c^2*d + d)*a*b*x + (c^3 + c)*a*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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