Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
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Rubi [A] time = 0.0185046, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-3),x]
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Rubi in Sympy [A] time = 1.7985, size = 19, normalized size = 0.79 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)**3,x)
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Mathematica [B] time = 0.085951, size = 71, normalized size = 2.96 \[ \frac{x \left (\left (2 n^2-3 n+1\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{a \left (a (3 n-1)+b (2 n-1) x^n\right )}{\left (a+b x^n\right )^2}\right )}{2 a^3 n^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-3),x]
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Maple [F] time = 0.073, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (2 \, n^{2} - 3 \, n + 1\right )} \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n - 1\right )} x x^{n} + a{\left (3 \, n - 1\right )} x}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-3),x, algorithm="fricas")
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Sympy [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(1/(a+b*x**n)**3,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-3),x, algorithm="giac")
[Out]