3.3.1 \(\int (c e+d e x) (a+b \sinh ^{-1}(c+d x))^{7/2} \, dx\) [201]

Optimal. Leaf size=305 \[ -\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {105 b^{7/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 b^{7/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d} \]

[Out]

35/64*b^2*e*(a+b*arcsinh(d*x+c))^(3/2)/d+35/32*b^2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(3/2)/d+1/4*e*(a+b*arcsinh
(d*x+c))^(7/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(7/2)/d-105/2048*b^(7/2)*e*exp(2*a/b)*erf(2^(1/2)*(a+b*a
rcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+105/2048*b^(7/2)*e*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)-7/8*b*e*(d*x+c)*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d-105/128*
b^3*e*(d*x+c)*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.53, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5859, 12, 5777, 5812, 5783, 5780, 5556, 3389, 2211, 2236, 2235} \begin {gather*} -\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}-\frac {105 b^3 e (c+d x) \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {7 b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(128*d) + (35*b^2*e*(a + b*ArcSinh[c
+ d*x])^(3/2))/(64*d) + (35*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(32*d) - (7*b*e*(c + d*x)*Sqrt[
1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(8*d) + (e*(a + b*ArcSinh[c + d*x])^(7/2))/(4*d) + (e*(c + d*
x)^2*(a + b*ArcSinh[c + d*x])^(7/2))/(2*d) - (105*b^(7/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*Arc
Sinh[c + d*x]])/Sqrt[b]])/(1024*d) + (105*b^(7/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqr
t[b]])/(1024*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*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {(7 b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {(7 b e) \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (35 b^2 e\right ) \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{16 d}\\ &=\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1024 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{512 d}+\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{512 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {105 b^{7/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 b^{7/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 125, normalized size = 0.41 \begin {gather*} \frac {e e^{-\frac {2 a}{b}} \left (b^4 \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {9}{2},-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+b^4 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {9}{2},\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{64 \sqrt {2} d \sqrt {a+b \sinh ^{-1}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(b^4*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + b^4*E^((4*a)/b)*Sqrt
[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(64*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*Arc
Sinh[c + d*x]])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right ) \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {7}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3876 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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