\(\int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{3/2} \, dx\) [189]

Optimal result

Integrand size = 23, antiderivative size = 205 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {3 b^{3/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {3 b^{3/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d} \]

[Out]

1/4*e*(a+b*arcsinh(d*x+c))^(3/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(3/2)/d-3/128*b^(3/2)*e*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+3/128*b^(3/2)*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)-3/8*b*e*(d*x+c)*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2
)/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5859, 12, 5777, 5812, 5783, 5780, 5556, 3389, 2211, 2236, 2235} \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {3 b e \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d} \]

[In]

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

[Out]

(-3*b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(8*d) + (e*(a + b*ArcSinh[c + d*x])^(3/2
))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqr
t[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d) + (3*b^(3/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh
[c + d*x]])/Sqrt[b]])/(64*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 x (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}+\frac {(3 b e) \text {Subst}\left (\int \frac {\sqrt {a+b \text {arcsinh}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{16 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \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 )}{16 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \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 )}{16 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \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 )}{32 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \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 )}{64 d}+\frac {(3 b e) \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 )}{64 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {(3 b e) \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 )}{32 d}+\frac {(3 b e) \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 )}{32 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d}-\frac {3 b^{3/2} e 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 )}{64 d}+\frac {3 b^{3/2} e 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 )}{64 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.61 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {e e^{-\frac {2 a}{b}} \left (b^2 \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {5}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+b^2 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{16 \sqrt {2} d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

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

[Out]

(e*(b^2*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + b^2*E^((4*a)/b)*Sqrt
[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(16*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*Arc
Sinh[c + d*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

e*(Integral(a*c*sqrt(a + b*asinh(c + d*x)), x) + Integral(a*d*x*sqrt(a + b*asinh(c + d*x)), x) + Integral(b*c*
sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), x) + Integral(b*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), x))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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