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

Optimal. Leaf size=124 \[ -\frac{2 c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

[Out]

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

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Rubi [A]  time = 0.219608, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.6104, size = 112, normalized size = 0.9 \[ - \frac{2 c x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} - \frac{2 c x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.03085, size = 261, normalized size = 2.1 \[ -2 c x \left (\frac{1-\left (\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-2^{-1/n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n+c*x^(2*n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(c*x^(2*n) + b*x^n + a), 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/(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. \[ \int \frac{1}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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