\(\int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx\) [217]

Optimal result

Integrand size = 25, antiderivative size = 326 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {e^3 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \]

[Out]

-2/3*e^3*exp(4*a/b)*erf(2*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d+2/3*e^3*erfi(2*(a+b*arcsinh(d
*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d/exp(4*a/b)+1/3*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)/b^(5/2)/d-1/3*e^3*erfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/
b^(5/2)/d/exp(2*a/b)-2/3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(3/2)-4*e^3*(d*x+c)^2/b^2/
d/(a+b*arcsinh(d*x+c))^(1/2)-16/3*e^3*(d*x+c)^4/b^2/d/(a+b*arcsinh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5859, 12, 5779, 5818, 5780, 5556, 3389, 2211, 2236, 2235} \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {\pi } e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {\sqrt {2 \pi } e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 \sqrt {\pi } e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {\sqrt {2 \pi } e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}} \]

[In]

Int[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

(-2*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e^3*(c + d*x)^2)/(b^2*d
*Sqrt[a + b*ArcSinh[c + d*x]]) - (16*e^3*(c + d*x)^4)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*e^3*E^((4*a)
/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (e^3*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(S
qrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c +
d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((4*a)/b)) - (e^3*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[
b]])/(3*b^(5/2)*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 5779

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n +
1)/Sqrt[1 + c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^
2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

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 5818

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Dist[f*(m/
(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && 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))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{b d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (64 e^3\right ) \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 )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (64 e^3\right ) \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 )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (8 e^3\right ) \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 )}{3 b^3 d}-\frac {\left (4 e^3\right ) \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 )}{b^3 d}+\frac {\left (16 e^3\right ) \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 )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (4 e^3\right ) \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 )}{3 b^3 d}+\frac {\left (4 e^3\right ) \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 )}{3 b^3 d}-\frac {\left (2 e^3\right ) \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 )}{b^3 d}+\frac {\left (2 e^3\right ) \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 )}{b^3 d}+\frac {\left (8 e^3\right ) \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 )}{3 b^3 d}-\frac {\left (8 e^3\right ) \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 )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (8 e^3\right ) \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 )}{3 b^3 d}+\frac {\left (8 e^3\right ) \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 )}{3 b^3 d}-\frac {\left (4 e^3\right ) \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 )}{b^3 d}+\frac {\left (4 e^3\right ) \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 )}{b^3 d}+\frac {\left (16 e^3\right ) \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 )}{3 b^3 d}-\frac {\left (16 e^3\right ) \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 )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 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 )}{3 b^{5/2} d}+\frac {e^3 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 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 )}{3 b^{5/2} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.20 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\frac {e^3 e^{-4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \left (-8 b e^{4 \text {arcsinh}(c+d x)} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+4 \sqrt {2} b e^{\frac {2 a}{b}+4 \text {arcsinh}(c+d x)} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} e^{\frac {4 a}{b}} \left (-\left (-1+e^{2 \text {arcsinh}(c+d x)}\right )^2 \left (b \left (-1+e^{4 \text {arcsinh}(c+d x)}\right )+8 a \left (1+e^{2 \text {arcsinh}(c+d x)}+e^{4 \text {arcsinh}(c+d x)}\right )+8 b \left (1+e^{2 \text {arcsinh}(c+d x)}+e^{4 \text {arcsinh}(c+d x)}\right ) \text {arcsinh}(c+d x)\right )-8 \sqrt {2} e^{\frac {2 a}{b}+4 \text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+16 e^{4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{12 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}} \]

[In]

Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

(e^3*(-8*b*E^(4*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]
))/b] + 4*Sqrt[2]*b*E^((2*a)/b + 4*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-2*(a +
b*ArcSinh[c + d*x]))/b] + (E^((4*a)/b)*(-((-1 + E^(2*ArcSinh[c + d*x]))^2*(b*(-1 + E^(4*ArcSinh[c + d*x])) +
8*a*(1 + E^(2*ArcSinh[c + d*x]) + E^(4*ArcSinh[c + d*x])) + 8*b*(1 + E^(2*ArcSinh[c + d*x]) + E^(4*ArcSinh[c +
d*x]))*ArcSinh[c + d*x])) - 8*Sqrt[2]*E^((2*a)/b + 4*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*Ar
cSinh[c + d*x])*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] + 16*E^(4*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcS
inh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b]))/2))/(12*b^2*d*E^(4*(a/b +
ArcSinh[c + d*x]))*(a + b*ArcSinh[c + d*x])^(3/2))

Maple [F]

\[\int \frac {\left (d e x +c e \right )^{3}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

[In]

int((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(5/2),x)

[Out]

int((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**3/(a+b*asinh(d*x+c))**(5/2),x)

[Out]

e**3*(Integral(c**3/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*
sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2), x) + Integral(d**3*x**3/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*
b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2), x) + Integra
l(3*c*d**2*x**2/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*sqrt
(a + b*asinh(c + d*x))*asinh(c + d*x)**2), x) + Integral(3*c**2*d*x/(a**2*sqrt(a + b*asinh(c + d*x)) + 2*a*b*s
qrt(a + b*asinh(c + d*x))*asinh(c + d*x) + b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3/(b*arcsinh(d*x + c) + a)^(5/2), x)

Giac [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3/(b*arcsinh(d*x + c) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(5/2),x)

[Out]

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(5/2), x)