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]

-2/3*(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)

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Rubi [A]  time = 0.38, antiderivative size = 296, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.40, size = 262, normalized size = 0.89 \[ -\frac {c \sqrt {a^2 c x^2+c} e^{-4 \sinh ^{-1}(a x)} \left (16 a^2 x^2 e^{4 \sinh ^{-1}(a x)}+64 a x \sqrt {a^2 x^2+1} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}+8 e^{8 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)-8 \sinh ^{-1}(a x)+16 e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )+16 e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )+1\right )}{24 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-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))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arcsinh \left (a x \right )^{\frac {5}{2}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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