\(\int \frac {(c+a^2 c x^2)^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx\) [507]

Optimal result

Integrand size = 23, antiderivative size = 296 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \]

[Out]

-2/3*(a^2*c*x^2+c)^(3/2)*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(3/2)+2/3*c*erf(2*arcsinh(a*x)^(1/2))*Pi^(1/2)*(a^2*
c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+2/3*c*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)
^(1/2)+2/3*c*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+2/3*c*er
fi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)-16/3*c*x*(a^2*x^2+1)*(
a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5790, 5814, 5791, 3393, 3388, 2211, 2235, 2236, 5819, 5556} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {2 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 c x \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}}{3 \sqrt {\text {arcsinh}(a x)}} \]

[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 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 3388

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 3393

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 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 5790

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[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)/(b*(n + 1)))*Simp[(d
+ e*x^2)^p/(1 + c^2*x^2)^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 5791

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

Rule 5814

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/(b*c*(
n + 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] - Dist[c*((m + 2*p + 1)/(b*f*(n + 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}, x] && EqQ[e, c^2*d]
&& LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 5819

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

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {c e^{-4 \text {arcsinh}(a x)} \sqrt {c+a^2 c x^2} \left (1+14 e^{4 \text {arcsinh}(a x)}+e^{8 \text {arcsinh}(a x)}+16 a^2 e^{4 \text {arcsinh}(a x)} x^2-8 \text {arcsinh}(a x)+8 e^{8 \text {arcsinh}(a x)} \text {arcsinh}(a x)+64 a e^{4 \text {arcsinh}(a x)} x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+16 e^{4 \text {arcsinh}(a x)} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+16 \sqrt {2} e^{4 \text {arcsinh}(a x)} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+16 \sqrt {2} e^{4 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )+16 e^{4 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )\right )}{24 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \]

[In]

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

[Out]

-1/24*(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*ArcSinh[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])*(-A
rcSinh[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
*ArcSinh[a*x]] + 16*E^(4*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 4*ArcSinh[a*x]]))/(a*E^(4*ArcSinh[a*x])*S
qrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))

Maple [F]

\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)/asinh(a*x)**(5/2), x)

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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