\(\int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\) [222]

Optimal result

Integrand size = 25, antiderivative size = 531 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}-\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}-\frac {9 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d} \]

[Out]

-16/15*e^4*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-4/3*e^4*(d*x+c)^5/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-1/30*
e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d+1/30*e^4*erfi((a+b*arcsinh(d*x+c))^(1/
2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d/exp(a/b)+9/20*e^4*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^
(1/2)*Pi^(1/2)/b^(7/2)/d-9/20*e^4*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/d/
exp(3*a/b)-5/12*e^4*exp(5*a/b)*erf(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(7/2)/d+5/12
*e^4*erfi(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(5*a/b)-2/5*e^4*(d*x+c)^4*
(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(5/2)-32/5*e^4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d
*x+c))^(1/2)-40/3*e^4*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5859, 12, 5779, 5818, 5778, 3389, 2211, 2236, 2235} \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {\sqrt {\pi } e^4 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 \sqrt {3 \pi } e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}-\frac {5 \sqrt {5 \pi } e^4 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}+\frac {\sqrt {\pi } e^4 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}-\frac {9 \sqrt {3 \pi } e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 \sqrt {5 \pi } e^4 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}-\frac {40 e^4 \sqrt {(c+d x)^2+1} (c+d x)^4}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {32 e^4 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 e^4 \sqrt {(c+d x)^2+1} (c+d x)^4}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}} \]

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(-2*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (16*e^4*(c + d*x)^3)/(15*b
^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e^4*(c + d*x)^5)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (32*e^4*
(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (40*e^4*(c + d*x)^4*Sqrt[1 + (c +
d*x)^2])/(3*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[
b]])/(30*b^(7/2)*d) + (9*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(20*b
^(7/2)*d) - (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d)
+ (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(30*b^(7/2)*d*E^(a/b)) - (9*e^4*Sqrt[3*Pi]*Erfi[(S
qrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(20*b^(7/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sq
rt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d*E^((5*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 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 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 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^4 x^4}{(a+b \text {arcsinh}(x))^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{(a+b \text {arcsinh}(x))^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {\left (8 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac {\left (2 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {\left (16 e^4\right ) \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}+\frac {\left (20 e^4\right ) \text {Subst}\left (\int \frac {x^4}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (32 e^4\right ) \text {Subst}\left (\int \left (-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}+\frac {\left (40 e^4\right ) \text {Subst}\left (\int \left (-\frac {5 \sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}+\frac {9 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}-\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (8 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 b^4 d}-\frac {\left (24 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}+\frac {\left (15 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b^4 d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (4 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}-\frac {\left (4 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 b^4 d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{6 b^4 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{12 b^4 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{12 b^4 d}-\frac {\left (12 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}+\frac {\left (12 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}+\frac {\left (15 e^4\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 b^4 d}-\frac {\left (15 e^4\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 b^4 d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (8 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}-\frac {\left (8 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^4 d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^4 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{6 b^4 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{6 b^4 d}-\frac {\left (24 e^4\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}+\frac {\left (24 e^4\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}+\frac {\left (15 e^4\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 b^4 d}-\frac {\left (15 e^4\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 b^4 d} \\ & = -\frac {2 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {16 e^4 (c+d x)^3}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^4 (c+d x)^5}{3 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {40 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}+\frac {9 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}-\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{30 b^{7/2} d}-\frac {9 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{20 b^{7/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{12 b^{7/2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.32 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {e^4 \left (-6 b^2 e^{\text {arcsinh}(c+d x)}-3 b^2 e^{5 \text {arcsinh}(c+d x)}+e^{-\text {arcsinh}(c+d x)} \left (-8 a^2+4 a b-6 b^2-4 (4 a-b) b \text {arcsinh}(c+d x)-8 b^2 \text {arcsinh}(c+d x)^2+8 e^{\frac {a}{b}+\text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x))^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-10 e^{-\frac {5 a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (10 a+b+10 b \text {arcsinh}(c+d x))+10 \sqrt {5} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )+9 \left (b^2 e^{3 \text {arcsinh}(c+d x)}+2 e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (6 a+b+6 b \text {arcsinh}(c+d x))+6 \sqrt {3} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )-4 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{\frac {a}{b}+\text {arcsinh}(c+d x)} (2 a+b+2 b \text {arcsinh}(c+d x))+2 b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )+9 e^{-3 \text {arcsinh}(c+d x)} \left (b^2+2 (a+b \text {arcsinh}(c+d x)) \left (6 a-b+6 b \text {arcsinh}(c+d x)-6 \sqrt {3} e^{3 \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 {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )-e^{-5 \text {arcsinh}(c+d x)} \left (3 b^2+10 (a+b \text {arcsinh}(c+d x)) \left (10 a-b+10 b \text {arcsinh}(c+d x)-10 \sqrt {5} e^{5 \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 {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )\right )}{240 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(e^4*(-6*b^2*E^ArcSinh[c + d*x] - 3*b^2*E^(5*ArcSinh[c + d*x]) + (-8*a^2 + 4*a*b - 6*b^2 - 4*(4*a - b)*b*ArcSi
nh[c + d*x] - 8*b^2*ArcSinh[c + d*x]^2 + 8*E^(a/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcS
inh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]])/E^ArcSinh[c + d*x] - (10*(a + b*ArcSinh[c + d*x])*(E^(5*(a
/b + ArcSinh[c + d*x]))*(10*a + b + 10*b*ArcSinh[c + d*x]) + 10*Sqrt[5]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2
)*Gamma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b]))/E^((5*a)/b) + 9*(b^2*E^(3*ArcSinh[c + d*x]) + (2*(a + b*ArcSin
h[c + d*x])*(E^(3*(a/b + ArcSinh[c + d*x]))*(6*a + b + 6*b*ArcSinh[c + d*x]) + 6*Sqrt[3]*b*(-((a + b*ArcSinh[c
+ d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b]))/E^((3*a)/b)) - (4*(a + b*ArcSinh[c + d*x])*(E
^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1
/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + (9*(b^2 + 2*(a + b*ArcSinh[c + d*x])*(6*a - b + 6*b*ArcSinh[c +
d*x] - 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1
/2, (3*(a + b*ArcSinh[c + d*x]))/b])))/E^(3*ArcSinh[c + d*x]) - (3*b^2 + 10*(a + b*ArcSinh[c + d*x])*(10*a - b
+ 10*b*ArcSinh[c + d*x] - 10*Sqrt[5]*E^(5*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSi
nh[c + d*x])*Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/E^(5*ArcSinh[c + d*x])))/(240*b^3*d*(a + b*ArcSinh[c
+ d*x])^(5/2))

Maple [F]

\[\int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]

[In]

int((d*e*x+c*e)^4/(a+b*arcsinh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)^4/(a+b*arcsinh(d*x+c))^(7/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsinh(d*x+c))^(7/2),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)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**4/(a+b*asinh(d*x+c))**(7/2),x)

[Out]

e**4*(Integral(c**4/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + 3*
a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x) +
Integral(d**4*x**4/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x) + 3*
a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3), x) +
Integral(4*c*d**3*x**3/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)
+ 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**3),
x) + Integral(6*c**2*d**2*x**2/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh(c + d*x))*asinh(c
+ d*x) + 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x
)**3), x) + Integral(4*c**3*d*x/(a**3*sqrt(a + b*asinh(c + d*x)) + 3*a**2*b*sqrt(a + b*asinh(c + d*x))*asinh(c
+ d*x) + 3*a*b**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)**2 + b**3*sqrt(a + b*asinh(c + d*x))*asinh(c + d*
x)**3), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^4/(b*arcsinh(d*x + c) + a)^(7/2), x)

Giac [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4/(b*arcsinh(d*x + c) + a)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]

[In]

int((c*e + d*e*x)^4/(a + b*asinh(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^4/(a + b*asinh(c + d*x))^(7/2), x)