Optimal. Leaf size=140 \[ \frac{c \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{x^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{c \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.125386, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{x^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{c \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^n + c*x^(2*n))),x]
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Rubi in Sympy [A] time = 24.295, size = 114, normalized size = 0.81 \[ \frac{c{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{x^{2} \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} + \frac{c{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{x^{2} \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b*x**n+c*x**(2*n)),x)
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Mathematica [A] time = 0.750385, size = 258, normalized size = 1.84 \[ \frac{c 2^{\frac{n+2}{n}} \left (\frac{\left (\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{2/n} \, _2F_1\left (\frac{n+2}{n},\frac{n+2}{n};2+\frac{2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}-b-2 c x^n}+\frac{x^{-n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{\frac{n+2}{n}} \, _2F_1\left (\frac{n+2}{n},\frac{n+2}{n};2+\frac{2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{c}\right )}{(n+2) x^2 \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^n + c*x^(2*n))),x]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b*x^n+c*x^(2*n)),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*x^3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{c x^{3} x^{2 \, n} + b x^{3} x^{n} + a x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*x^3),x, algorithm="fricas")
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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/x**3/(a+b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*x^3),x, algorithm="giac")
[Out]