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 = 0.146122, 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 \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2} \]

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)

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Rubi in Sympy [A]  time = 10.6509, size = 51, normalized size = 0.84 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + b x^{n}}} \right )}}{n + 2} + \frac{2 x \sqrt{\frac{a}{x^{2}} + b x^{n}}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a/x**2+b*x**n)**(1/2),x)

[Out]

-2*sqrt(a)*atanh(sqrt(a)/(x*sqrt(a/x**2 + b*x**n)))/(n + 2) + 2*x*sqrt(a/x**2 +
b*x**n)/(n + 2)

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Mathematica [A]  time = 0.147797, size = 95, normalized size = 1.56 \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n} \left (\sqrt{a+b x^{n+2}}-\sqrt{a} \log \left (\sqrt{a} \sqrt{a+b x^{n+2}}+a\right )+\sqrt{a} \log \left (x^{\frac{n}{2}+1}\right )\right )}{(n+2) \sqrt{a+b x^{n+2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x^2 + b*x^n],x]

[Out]

(2*x*Sqrt[a/x^2 + b*x^n]*(Sqrt[a + b*x^(2 + n)] + Sqrt[a]*Log[x^(1 + n/2)] - Sqr
t[a]*Log[a + Sqrt[a]*Sqrt[a + b*x^(2 + n)]]))/((2 + n)*Sqrt[a + b*x^(2 + n)])

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int \sqrt{{\frac{a}{{x}^{2}}}+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a/x^2+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a/x^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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a/x^2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + \frac{a}{x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a/x^2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a/x^2), x)