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

Optimal. Leaf size=61 \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2} \]

[Out]

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

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Rubi [A]  time = 0.0834905, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2007, 2029, 206} \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^2 + b*x^n],x]

[Out]

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

Rule 2007

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(p*(n - j)), x] + Dist
[a, Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[
Simplify[j*p + 1], 0]

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

Mathematica [A]  time = 0.0532376, size = 78, normalized size = 1.28 \[ \frac{x \sqrt{\frac{a}{x^2}+b x^n} \left (2 \sqrt{a+b x^{n+2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+2}}}{\sqrt{a}}\right )\right )}{(n+2) \sqrt{a+b x^{n+2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^2 + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.34, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\frac{a}{{x}^{2}}}+b{x}^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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*} \int \sqrt{b x^{n} + \frac{a}{x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^n + a/x^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/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*} \int \sqrt{\frac{a}{x^{2}} + b x^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x**2 + b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^n + a/x^2), x)