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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]