Optimal. Leaf size=147 \[ \frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]
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Rubi [A] time = 0.236429, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-((1 + 4*n)/n)),x]
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Rubi in Sympy [A] time = 4.62421, size = 46, normalized size = 0.31 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{4 + \frac{1}{n}} \left (a + b x^{n}\right )^{-4 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} 4 + \frac{1}{n}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((a+b*x**n)**((1+4*n)/n)),x)
[Out]
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Mathematica [C] time = 0.0712452, size = 55, normalized size = 0.37 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-((1 + 4*n)/n)),x]
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Maple [F] time = 0.17, size = 0, normalized size = 0. \[ \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1+4\,n}{n}}} \right ) ^{-1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((a+b*x^n)^((1+4*n)/n)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{4 \, n + 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((4*n + 1)/n)),x, algorithm="maxima")
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Fricas [A] time = 0.239461, size = 259, normalized size = 1.76 \[ \frac{6 \, b^{4} n^{3} x x^{4 \, n} + 6 \,{\left (4 \, a b^{3} n^{3} + a b^{3} n^{2}\right )} x x^{3 \, n} + 3 \,{\left (12 \, a^{2} b^{2} n^{3} + 7 \, a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x x^{2 \, n} +{\left (24 \, a^{3} b n^{3} + 26 \, a^{3} b n^{2} + 9 \, a^{3} b n + a^{3} b\right )} x x^{n} +{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )} x}{{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )}{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((4*n + 1)/n)),x, algorithm="fricas")
[Out]
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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+4*n)/n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^((4*n + 1)/n)),x, algorithm="giac")
[Out]