Optimal. Leaf size=56 \[ \frac{x^{4-n} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{8}{n}-1\right );\frac{4}{n};-\frac{b x^n}{a}\right )}{a (4-n)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0751198, antiderivative size = 65, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{4-n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n}-1;\frac{4}{n};-\frac{b x^n}{a}\right )}{(4-n) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x^(3 - n)/Sqrt[a + b*x^n],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.02197, size = 49, normalized size = 0.88 \[ \frac{x^{- n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{n - 4}{n} \\ \frac{4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (- n + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3-n)/(a+b*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.24922, size = 83, normalized size = 1.48 \[ \frac{x^4 \left (-b (n-8) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )-8 a x^{-n}-8 b\right )}{8 a (n-4) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3 - n)/Sqrt[a + b*x^n],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int{{x}^{3-n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3-n)/(a+b*x^n)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-n + 3)/sqrt(b*x^n + a),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(x^(-n + 3)/sqrt(b*x^n + a),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(x**(3-n)/(a+b*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-n + 3)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]