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

Optimal. Leaf size=42 \[ -\frac{2 \sqrt{a^2 x^2+1}}{3 a \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))

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Rubi [A]  time = 0.070311, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5677, 5675} \[ -\frac{2 \sqrt{a^2 x^2+1}}{3 a \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))

Rule 5677

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.039566, size = 42, normalized size = 1. \[ -\frac{2 \sqrt{a^2 x^2+1}}{3 a \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))

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Maple [A]  time = 0.042, size = 36, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,a}\sqrt{{a}^{2}{x}^{2}+1} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/arcsinh(a*x)^(3/2)/a/(c*(a^2*x^2+1))^(1/2)*(a^2*x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.49659, size = 134, normalized size = 3.19 \begin{align*} -\frac{2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1}}{3 \,{\left (a^{3} c x^{2} + a c\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)/((a^3*c*x^2 + a*c)*log(a*x + sqrt(a^2*x^2 + 1))^(3/2))

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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(1/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 \frac{1}{\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(1/arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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