3.566 \(\int \frac{1}{x^2 \left (a+b x^n+c x^{2 n}\right )} \, dx\)

Optimal. Leaf size=142 \[ \frac{2 c \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{x \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

[Out]

(2*c*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*
c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*x) + (2*c*Hypergeometric2F1[1, -n^(-1
), -((1 - n)/n), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2
- 4*a*c])*x)

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Rubi [A]  time = 0.139895, antiderivative size = 142, 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 \[ \frac{2 c \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{x \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{x \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^n + c*x^(2*n))),x]

[Out]

(2*c*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*
c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*x) + (2*c*Hypergeometric2F1[1, -n^(-1
), -((1 - n)/n), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2
- 4*a*c])*x)

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Rubi in Sympy [A]  time = 24.3419, size = 114, normalized size = 0.8 \[ \frac{2 c{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{x \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} + \frac{2 c{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{x \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(a+b*x**n+c*x**(2*n)),x)

[Out]

2*c*hyper((1, -1/n), ((n - 1)/n,), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(x*(-4*a
*c + b**2 + b*sqrt(-4*a*c + b**2))) + 2*c*hyper((1, -1/n), ((n - 1)/n,), -2*c*x*
*n/(b - sqrt(-4*a*c + b**2)))/(x*(-4*a*c + b**2 - b*sqrt(-4*a*c + b**2)))

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Mathematica [A]  time = 0.726893, size = 240, normalized size = 1.69 \[ \frac{c 2^{\frac{1}{n}+1} \left (\frac{\left (\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{\frac{1}{n}} \, _2F_1\left (1+\frac{1}{n},1+\frac{1}{n};2+\frac{1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}-b-2 c x^n}+\frac{x^{-n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{\frac{1}{n}+1} \, _2F_1\left (1+\frac{1}{n},1+\frac{1}{n};2+\frac{1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{c}\right )}{(n+1) x \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^n + c*x^(2*n))),x]

[Out]

(2^(1 + n^(-1))*c*((((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)*Hypergeom
etric2F1[1 + n^(-1), 1 + n^(-1), 2 + n^(-1), (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b
^2 - 4*a*c] + 2*c*x^n)])/(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n) + (((c*x^n)/(b + Sqr
t[b^2 - 4*a*c] + 2*c*x^n))^(1 + n^(-1))*Hypergeometric2F1[1 + n^(-1), 1 + n^(-1)
, 2 + n^(-1), (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(c*x^n
)))/(Sqrt[b^2 - 4*a*c]*(1 + n)*x)

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n+c*x^(2*n)),x)

[Out]

int(1/x^2/(a+b*x^n+c*x^(2*n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{c x^{2} x^{2 \, n} + b x^{2} x^{n} + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="fricas")

[Out]

integral(1/(c*x^2*x^(2*n) + b*x^2*x^n + a*x^2), x)

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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**2/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)*x^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)*x^2), x)