3.2497 \(\int \frac{1}{x \sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

[Out]

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

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Rubi [A]  time = 0.0528606, antiderivative size = 28, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*x^n]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.72062, size = 26, normalized size = 0.93 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{\sqrt{a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.024119, size = 28, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 23, normalized size = 0.8 \[ -2\,{\frac{1}{n\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n)^(1/2),x)

[Out]

-2*arctanh((a+b*x^n)^(1/2)/a^(1/2))/n/a^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227175, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right )}{\sqrt{a} n}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right )}{\sqrt{-a} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x),x, algorithm="fricas")

[Out]

[log((sqrt(a)*b*x^n - 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^n)/(sqrt(a)*n), 2*arcta
n(a/(sqrt(b*x^n + a)*sqrt(-a)))/(sqrt(-a)*n)]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n + a)*x), x)