3.507 \(\int \frac{(c+a^2 c x^2)^{3/2}}{\sinh ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=296 \[ \frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(3*a*ArcSinh[a*x]^(3/2)) - (16*c*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x
^2])/(3*Sqrt[ArcSinh[a*x]]) + (2*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x
^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqr
t[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c
*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2])

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Rubi [A]  time = 0.376707, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5696, 5777, 5699, 3312, 3307, 2180, 2204, 2205, 5779, 5448} \[ \frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{3 a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(5/2),x]

[Out]

(-2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(3*a*ArcSinh[a*x]^(3/2)) - (16*c*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x
^2])/(3*Sqrt[ArcSinh[a*x]]) + (2*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x
^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqr
t[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c
*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2])

Rule 5696

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

Rule 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntP
art[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p -
1/2)*(a + b*ArcSinh[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(
n + 1)*(1 + c^2*x^2)^FracPart[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}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]

Rule 5699

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[
(a + b*x)^n*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IG
tQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

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]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 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]

Rubi steps

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

Mathematica [A]  time = 0.396522, size = 262, normalized size = 0.89 \[ -\frac{c \sqrt{a^2 c x^2+c} e^{-4 \sinh ^{-1}(a x)} \left (16 e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 \sinh ^{-1}(a x)\right )+16 \sqrt{2} e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )+16 \sqrt{2} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )+16 e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},4 \sinh ^{-1}(a x)\right )+16 a^2 x^2 e^{4 \sinh ^{-1}(a x)}+64 a x \sqrt{a^2 x^2+1} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}+8 e^{8 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)-8 \sinh ^{-1}(a x)+1\right )}{24 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(5/2),x]

[Out]

-(c*Sqrt[c + a^2*c*x^2]*(1 + 14*E^(4*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) + 16*a^2*E^(4*ArcSinh[a*x])*x^2 - 8*Ar
cSinh[a*x] + 8*E^(8*ArcSinh[a*x])*ArcSinh[a*x] + 64*a*E^(4*ArcSinh[a*x])*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + 16
*E^(4*ArcSinh[a*x])*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -4*ArcSinh[a*x]] + 16*Sqrt[2]*E^(4*ArcSinh[a*x])*(-ArcSin
h[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + 16*Sqrt[2]*E^(4*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcS
inh[a*x]] + 16*E^(4*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 4*ArcSinh[a*x]]))/(24*a*E^(4*ArcSinh[a*x])*Sqr
t[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))

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Maple [F]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),x)

[Out]

int((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\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((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)/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((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),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((a**2*c*x**2+c)**(3/2)/asinh(a*x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\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((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)/arcsinh(a*x)^(5/2), x)