3.396 \(\int \frac{1}{c^2 x^2 \sqrt{\frac{a}{x^2}+b x^n}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{\sqrt{a} c^2 (n+2)} \]

[Out]

(-2*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(Sqrt[a]*c^2*(2 + n))

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Rubi [A]  time = 0.0737836, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {12, 2029, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{\sqrt{a} c^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(Sqrt[a]*c^2*(2 + n))

Rule 12

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

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{c^2 x^2 \sqrt{\frac{a}{x^2}+b x^n}} \, dx &=\frac{\int \frac{1}{x^2 \sqrt{\frac{a}{x^2}+b x^n}} \, dx}{c^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{c^2 (2+n)}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{\sqrt{a} c^2 (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0466403, size = 66, normalized size = 1.65 \[ -\frac{2 \sqrt{a+b x^{n+2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+2}}}{\sqrt{a}}\right )}{\sqrt{a} c^2 (n+2) x \sqrt{\frac{a}{x^2}+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^(2 + n)]*ArcTanh[Sqrt[a + b*x^(2 + n)]/Sqrt[a]])/(Sqrt[a]*c^2*(2 + n)*x*Sqrt[a/x^2 + b*x^n])

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Maple [F]  time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{a}{{x}^{2}}}+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/c^2/x^2/(1/x^2*a+b*x^n)^(1/2),x)

[Out]

int(1/c^2/x^2/(1/x^2*a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{b x^{n} + \frac{a}{x^{2}}} x^{2}}\,{d x}}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a/x^2)*x^2), x)/c^2

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{2} \sqrt{\frac{a}{x^{2}} + b x^{n}}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c**2/x**2/(a/x**2+b*x**n)**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(a/x**2 + b*x**n)), x)/c**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x^{2}}} c^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a/x^2)*c^2*x^2), x)