Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n-b x^2}}\right )}{\sqrt{b} (2-n)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0410929, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n-b x^2}}\right )}{\sqrt{b} (2-n)} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[x^n*(a - b*x^(2 - n))],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.25418, size = 31, normalized size = 0.82 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a x^{n} - b x^{2}}} \right )}}{\sqrt{b} \left (- n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**n*(a-b*x**(2-n)))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.128499, size = 80, normalized size = 2.11 \[ -\frac{2 \sqrt{a} x^{n/2} \sqrt{1-\frac{b x^{2-n}}{a}} \sin ^{-1}\left (\frac{\sqrt{b} x^{1-\frac{n}{2}}}{\sqrt{a}}\right )}{\sqrt{b} (n-2) \sqrt{a x^n-b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[x^n*(a - b*x^(2 - n))],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{x}^{n} \left ( a-b{x}^{2-n} \right ) }}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^n*(a-b*x^(2-n)))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (b x^{-n + 2} - a\right )} x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(b*x^(-n + 2) - a)*x^n),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
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(-(b*x^(-n + 2) - a)*x^n),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**n*(a-b*x**(2-n)))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (b x^{-n + 2} - a\right )} x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(b*x^(-n + 2) - a)*x^n),x, algorithm="giac")
[Out]