\(\int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) [205]

Optimal result

Integrand size = 25, antiderivative size = 217 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \]

[Out]

1/16*e^3*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/b^(1/2)-1/16*e^3*erfi(2
^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)/b^(1/2)-1/32*e^3*exp(4*a/b)*erf(2*(a+
b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/b^(1/2)+1/32*e^3*erfi(2*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/
2)/d/exp(4*a/b)/b^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5859, 12, 5780, 5556, 3389, 2211, 2236, 2235} \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {\sqrt {\pi } e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {\sqrt {\pi } e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {\sqrt {\frac {\pi }{2}} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \]

[In]

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

[Out]

-1/32*(e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d) + (e^3*E^((2*a)/b)*
Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a +
b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((4*a)/b)) - (e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[
c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d*E^((2*a)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 \frac {e^3 x^3}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = -\frac {e^3 \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = -\frac {e^3 \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b d}+\frac {e^3 \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 b d} \\ & = -\frac {e^3 \text {Subst}\left (\int \frac {e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b d}+\frac {e^3 \text {Subst}\left (\int \frac {e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{16 b d}+\frac {e^3 \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b d}-\frac {e^3 \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b d} \\ & = -\frac {e^3 \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 b d}+\frac {e^3 \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 b d}+\frac {e^3 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 b d}-\frac {e^3 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 b d} \\ & = -\frac {e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-2 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \left (-2 \sqrt {2} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{32 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

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

[Out]

(e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - 2*Sqrt[2]*E^((2*a)/b)*
Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[a/b + ArcSi
nh[c + d*x]]*(-2*Sqrt[2]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[1/2, (4*(a + b*ArcSinh
[c + d*x]))/b])))/(32*d*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])

Maple [F]

\[\int \frac {\left (d e x +c e \right )^{3}}{\sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

integrate((d*e*x+c*e)**3/(a+b*asinh(d*x+c))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(1/2),x)

[Out]

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(1/2), x)