\(\int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx\) [199]

Optimal result

Integrand size = 25, antiderivative size = 547 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {105 b^{7/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{131072 d}+\frac {105 b^{7/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {105 b^{7/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{131072 d}-\frac {105 b^{7/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d} \]

[Out]

-525/2048*b^2*e^3*(a+b*arcsinh(d*x+c))^(3/2)/d-105/256*b^2*e^3*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(3/2)/d+35/256*b
^2*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^(3/2)/d-3/32*e^3*(a+b*arcsinh(d*x+c))^(7/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arc
sinh(d*x+c))^(7/2)/d+105/4096*b^(7/2)*e^3*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*P
i^(1/2)/d-105/4096*b^(7/2)*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)-
105/131072*b^(7/2)*e^3*exp(4*a/b)*erf(2*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d+105/131072*b^(7/2)*e^3*
erfi(2*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(4*a/b)+21/64*b*e^3*(d*x+c)*(a+b*arcsinh(d*x+c))^(5/2
)*(1+(d*x+c)^2)^(1/2)/d-7/32*b*e^3*(d*x+c)^3*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d+1575/4096*b^3*e^
3*(d*x+c)*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d-105/2048*b^3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)*(a+b
*arcsinh(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5859, 12, 5777, 5812, 5783, 5780, 5556, 3389, 2211, 2236, 2235} \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=-\frac {105 \sqrt {\pi } b^{7/2} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{131072 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {105 \sqrt {\pi } b^{7/2} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{131072 d}-\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {105 b^3 e^3 \sqrt {(c+d x)^2+1} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}+\frac {1575 b^3 e^3 \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {7 b e^3 \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}+\frac {21 b e^3 \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d} \]

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(1575*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(4096*d) - (105*b^3*e^3*(c + d*x)^
3*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2048*d) - (525*b^2*e^3*(a + b*ArcSinh[c + d*x])^(3/2))/
(2048*d) - (105*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(256*d) + (35*b^2*e^3*(c + d*x)^4*(a + b*A
rcSinh[c + d*x])^(3/2))/(256*d) + (21*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(6
4*d) - (7*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(32*d) - (3*e^3*(a + b*ArcSi
nh[c + d*x])^(7/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^(7/2))/(4*d) - (105*b^(7/2)*e^3*E^((4*a
)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(131072*d) + (105*b^(7/2)*e^3*E^((2*a)/b)*Sqrt[Pi
/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2048*d) + (105*b^(7/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a +
b*ArcSinh[c + d*x]])/Sqrt[b]])/(131072*d*E^((4*a)/b)) - (105*b^(7/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*
ArcSinh[c + d*x]])/Sqrt[b]])/(2048*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 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 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] && GtQ[n, 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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -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^3 x^3 (a+b \text {arcsinh}(x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (7 b e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}+\frac {\left (21 b e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (35 b^2 e^3\right ) \text {Subst}\left (\int x^3 (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{64 d} \\ & = \frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (21 b e^3\right ) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (105 b^2 e^3\right ) \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{128 d}-\frac {\left (105 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{512 d} \\ & = -\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}+\frac {\left (315 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2048 d}+\frac {\left (315 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{512 d}+\frac {\left (105 b^4 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{4096 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (105 b^3 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 )}{4096 d}-\frac {\left (315 b^3 e^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4096 d}-\frac {\left (315 b^3 e^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{1024 d}-\frac {\left (315 b^4 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{8192 d}-\frac {\left (315 b^4 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{2048 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (105 b^3 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 )}{4096 d}+\frac {\left (315 b^3 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 )}{8192 d}+\frac {\left (315 b^3 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 )}{2048 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (105 b^3 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 )}{32768 d}+\frac {\left (105 b^3 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 )}{16384 d}+\frac {\left (315 b^3 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 )}{8192 d}+\frac {\left (315 b^3 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 )}{2048 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (105 b^3 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 )}{65536 d}+\frac {\left (105 b^3 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 )}{65536 d}+\frac {\left (105 b^3 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 )}{32768 d}-\frac {\left (105 b^3 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 )}{32768 d}+\frac {\left (315 b^3 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 )}{16384 d}+\frac {\left (315 b^3 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 )}{4096 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {\left (105 b^3 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 )}{32768 d}+\frac {\left (105 b^3 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 )}{32768 d}+\frac {\left (105 b^3 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 )}{16384 d}-\frac {\left (105 b^3 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 )}{16384 d}+\frac {\left (315 b^3 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 )}{32768 d}-\frac {\left (315 b^3 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 )}{32768 d}+\frac {\left (315 b^3 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 )}{8192 d}-\frac {\left (315 b^3 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 )}{8192 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {105 b^{7/2} 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 )}{131072 d}+\frac {105 b^{7/2} 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 )}{32768 d}+\frac {105 b^{7/2} 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 )}{131072 d}-\frac {105 b^{7/2} 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 )}{32768 d}+\frac {\left (315 b^3 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 )}{16384 d}-\frac {\left (315 b^3 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 )}{16384 d}+\frac {\left (315 b^3 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 )}{4096 d}-\frac {\left (315 b^3 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 )}{4096 d} \\ & = \frac {1575 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{4096 d}-\frac {105 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2048 d}-\frac {525 b^2 e^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{2048 d}-\frac {105 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {35 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{3/2}}{256 d}+\frac {21 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{64 d}-\frac {7 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{32 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^{7/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^{7/2}}{4 d}-\frac {105 b^{7/2} 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 )}{131072 d}+\frac {105 b^{7/2} 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 )}{2048 d}+\frac {105 b^{7/2} 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 )}{131072 d}-\frac {105 b^{7/2} 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 )}{2048 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.38 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {b^4 e^3 e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-32 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \left (-32 \sqrt {2} \Gamma \left (\frac {9}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {9}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{8192 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(7/2),x]

[Out]

(b^4*e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - 32*Sqrt[2]*E^((2*a
)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[a/b +
ArcSinh[c + d*x]]*(-32*Sqrt[2]*Gamma[9/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[9/2, (4*(a + b*A
rcSinh[c + d*x]))/b])))/(8192*d*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])

Maple [F]

\[\int \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]

[In]

int((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^(7/2),x)

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^3*(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(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((d*e*x+c*e)**3*(a+b*asinh(d*x+c))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3*(b*arcsinh(d*x + c) + a)^(7/2), x)

Giac [F]

\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3*(b*arcsinh(d*x + c) + a)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]

[In]

int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^(7/2), x)