Optimal. Leaf size=97 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.123963, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-((1 + 3*n)/n)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.66019, size = 46, normalized size = 0.47 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{3 + \frac{1}{n}} \left (a + b x^{n}\right )^{-3 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} 3 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((a+b*x**n)**((1+3*n)/n)),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0465988, size = 55, normalized size = 0.57 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-((1 + 3*n)/n)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.149, size = 0, normalized size = 0. \[ \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1+3\,n}{n}}} \right ) ^{-1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((a+b*x^n)^((1+3*n)/n)),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{3 \, n + 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((3*n + 1)/n)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.238664, size = 170, normalized size = 1.75 \[ \frac{2 \, b^{3} n^{2} x x^{3 \, n} + 2 \,{\left (3 \, a b^{2} n^{2} + a b^{2} n\right )} x x^{2 \, n} +{\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} +{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} x}{{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )}{\left (b x^{n} + a\right )}^{\frac{3 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((3*n + 1)/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/((a+b*x**n)**((1+3*n)/n)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((3*n + 1)/n)),x, algorithm="giac")
[Out]