Optimal. Leaf size=34 \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]
[Out]
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Rubi [A] time = 0.029681, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^n)^3),x]
[Out]
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Rubi in Sympy [A] time = 3.66769, size = 22, normalized size = 0.65 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} 3, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*x**n)**3,x)
[Out]
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Mathematica [B] time = 0.0996309, size = 76, normalized size = 2.24 \[ \frac{\frac{a \left (3 a n+a+b (2 n+1) x^n\right )}{\left (a+b x^n\right )^2}-\left (2 n^2+3 n+1\right ) \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{2 a^3 n^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^n)^3),x]
[Out]
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Maple [F] time = 0.094, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*x^n)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (2 \, n^{2} + 3 \, n + 1\right )} \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{2} x^{n} + a^{3} n^{2} x^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n + 1\right )} x^{n} + a{\left (3 \, n + 1\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x x^{2 \, n} + 2 \, a^{3} b n^{2} x x^{n} + a^{4} n^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^3*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{3} x^{2} x^{3 \, n} + 3 \, a b^{2} x^{2} x^{2 \, n} + 3 \, a^{2} b x^{2} x^{n} + a^{3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^3*x^2),x, algorithm="fricas")
[Out]
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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/x**2/(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^3*x^2),x, algorithm="giac")
[Out]