Optimal. Leaf size=96 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]
[Out]
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Rubi [A] time = 0.0949738, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-3 - n^(-1)),x]
[Out]
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Rubi in Sympy [A] time = 10.6053, size = 80, normalized size = 0.83 \[ \frac{x \left (a + b x^{n}\right )^{-2 - \frac{1}{n}}}{a \left (2 n + 1\right )} + \frac{2 n x \left (a + b x^{n}\right )^{-1 - \frac{1}{n}}}{a^{2} \left (n + 1\right ) \left (2 n + 1\right )} + \frac{2 n^{2} x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a^{3} \left (n + 1\right ) \left (2 n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(-3-1/n),x)
[Out]
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Mathematica [C] time = 0.0513605, 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)^(-3 - n^(-1)),x]
[Out]
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Maple [F] time = 0.15, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-3-{n}^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(-3-1/n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242118, size = 170, normalized size = 1.77 \[ \frac{2 \, b^{3} n^{2} x x^{3 \, n} + 2 \,{\left (3 \, a b^{2} n^{2} + a b^{2} n\right )} x x^{2 \, n} +{\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} +{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} x}{{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )}{\left (b x^{n} + a\right )}^{\frac{3 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 3),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((a+b*x**n)**(-3-1/n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-1/n - 3),x, algorithm="giac")
[Out]