3.2683 \(\int \frac{x^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} n} \]

[Out]

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

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Rubi [A]  time = 0.0488729, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.30955, size = 29, normalized size = 0.83 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{\sqrt{b} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0302835, size = 38, normalized size = 1.09 \[ \frac{2 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )}{\sqrt{b} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{1{x}^{-1+{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+1/2*n)/(a+b*x^n)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: TypeError

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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(x**(-1+1/2*n)/(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. \[ \int \frac{x^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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