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

Optimal. Leaf size=37 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{\sqrt{a} (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.033683, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{\sqrt{a} (2-n)} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a*x^2 + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.16503, size = 31, normalized size = 0.84 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{n}}} \right )}}{\sqrt{a} \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.101691, size = 78, normalized size = 2.11 \[ -\frac{2 \sqrt{b} x^{n/2} \sqrt{\frac{a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{1-\frac{n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} (n-2) \sqrt{a x^2+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a*x^2 + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{a{x}^{2}+b{x}^{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^2+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a*x**2 + b*x**n), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{2} + b x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(a*x^2 + b*x^n), x)