Optimal. Leaf size=67 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.074149, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.2043, size = 61, normalized size = 0.91 \[ - \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 3, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} x^{2} \left (a b + b^{2} x^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.110042, size = 94, normalized size = 1.4 \[ \frac{\left (a+b x^n\right ) \left (a^2 n-\left (n^2+3 n+2\right ) \left (a+b x^n\right )^2 \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )+2 a (n+1) \left (a+b x^n\right )\right )}{2 a^3 n^2 x^2 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (n^{2} + 3 \, n + 2\right )} \int \frac{1}{a^{2} b n^{2} x^{3} x^{n} + a^{3} n^{2} x^{3}}\,{d x} + \frac{2 \, b{\left (n + 1\right )} x^{n} + a{\left (3 \, n + 2\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{2} x^{n} + a^{4} n^{2} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{3} x^{2 \, n} + 2 \, a b x^{3} x^{n} + a^{2} x^{3}\right )} \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),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**3/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^3),x, algorithm="giac")
[Out]