3.482 \(\int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=298 \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{a^2 x^2+1}}-\frac{5 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{a^2 x^2+1}}+\frac{15}{32} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]

[Out]

(15*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/32 - (5*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(16*a*Sqrt[1 + a
^2*x^2]) - (5*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(8*Sqrt[1 + a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*Arc
Sinh[a*x]^(5/2))/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[1 + a^2*x^2]) + (15*Sqrt[Pi/2]*Sqrt[c
+ a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) - (15*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*E
rfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2])

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Rubi [A]  time = 0.300659, antiderivative size = 298, 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 = {5682, 5675, 5663, 5758, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{a^2 x^2+1}}-\frac{5 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{a^2 x^2+1}}+\frac{15}{32} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2),x]

[Out]

(15*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/32 - (5*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(16*a*Sqrt[1 + a
^2*x^2]) - (5*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(8*Sqrt[1 + a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*Arc
Sinh[a*x]^(5/2))/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[1 + a^2*x^2]) + (15*Sqrt[Pi/2]*Sqrt[c
+ a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) - (15*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*E
rfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2])

Rule 5682

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

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 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 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 12

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

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 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \int \frac{\sinh ^{-1}(a x)^{5/2}}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{1+a^2 x^2}}-\frac{\left (5 a \sqrt{c+a^2 c x^2}\right ) \int x \sinh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt{1+a^2 x^2}}\\ &=-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}+\frac{\left (15 a^2 \sqrt{c+a^2 c x^2}\right ) \int \frac{x^2 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{16 \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{32 \sqrt{1+a^2 x^2}}-\frac{\left (15 a \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{64 \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}+\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}+\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{128 a \sqrt{1+a^2 x^2}}-\frac{\left (15 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{128 a \sqrt{1+a^2 x^2}}\\ &=\frac{15}{32} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{5 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{1+a^2 x^2}}-\frac{5 a x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{1+a^2 x^2}}+\frac{1}{2} x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{1+a^2 x^2}}+\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.290943, size = 135, normalized size = 0.45 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (105 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )-105 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+8 \sqrt{\sinh ^{-1}(a x)} \left (64 \sinh ^{-1}(a x)^3+7 \left (16 \sinh ^{-1}(a x)^2+15\right ) \sinh \left (2 \sinh ^{-1}(a x)\right )-140 \sinh ^{-1}(a x) \cosh \left (2 \sinh ^{-1}(a x)\right )\right )\right )}{3584 a \sqrt{a^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2),x]

[Out]

(Sqrt[c*(1 + a^2*x^2)]*(105*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 105*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcS
inh[a*x]]] + 8*Sqrt[ArcSinh[a*x]]*(64*ArcSinh[a*x]^3 - 140*ArcSinh[a*x]*Cosh[2*ArcSinh[a*x]] + 7*(15 + 16*ArcS
inh[a*x]^2)*Sinh[2*ArcSinh[a*x]])))/(3584*a*Sqrt[1 + a^2*x^2])

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{a}^{2}c{x}^{2}+c}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^(5/2), x)

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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(arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(5/2)*(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^(5/2), x)