3.189 \(\int (c e+d e x) (a+b \sinh ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=205 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b e \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.482889, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5865, 12, 5663, 5758, 5675, 5669, 5448, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b e \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[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 5865

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
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 5663

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 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*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}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

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 3308

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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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]

Rubi steps

\begin{align*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(c+d x)}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(3 b e) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sinh ^{-1}(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 \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{3 b e (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{3 b e (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 d}\\ &=-\frac{3 b e (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac{3 b e (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32 d}+\frac{(3 b e) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32 d}\\ &=-\frac{3 b e (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{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 \sinh ^{-1}(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 \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}\\ \end{align*}

Mathematica [A]  time = 0.104699, size = 142, normalized size = 0.69 \[ \frac{b e e^{-\frac{2 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{5}{2},\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{5}{2},-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{16 \sqrt{2} d \sqrt{-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.137, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)