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

Optimal. Leaf size=140 \[ -\frac{2 c x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

[Out]

(-2*c*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]
)])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (2*c*x^3*Hypergeometric2F1[1, 3/n,
(3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 -
4*a*c]))

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Rubi [A]  time = 0.28012, 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 \[ -\frac{2 c x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]
)])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (2*c*x^3*Hypergeometric2F1[1, 3/n,
(3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 -
4*a*c]))

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Rubi in Sympy [A]  time = 24.611, size = 122, normalized size = 0.87 \[ - \frac{2 c x^{3}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{3}{n} \\ \frac{n + 3}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{3 \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} - \frac{2 c x^{3}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{3}{n} \\ \frac{n + 3}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{3 \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*x**3*hyper((1, 3/n), ((n + 3)/n,), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(3*
(-4*a*c + b**2 + b*sqrt(-4*a*c + b**2))) - 2*c*x**3*hyper((1, 3/n), ((n + 3)/n,)
, -2*c*x**n/(b - sqrt(-4*a*c + b**2)))/(3*(-4*a*c + b**2 - b*sqrt(-4*a*c + b**2)
))

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Mathematica [A]  time = 1.16769, size = 265, normalized size = 1.89 \[ -\frac{2}{3} c x^3 \left (\frac{1-\left (\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right )^{-3/n} \, _2F_1\left (-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-8^{-1/n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-3/n} \, _2F_1\left (-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*x^3*((1 - Hypergeometric2F1[-3/n, -3/n, (-3 + 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
))^(3/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-3/n, -3/
n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(
-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(3/n)))/(Sqrt[b^2 - 4*a*c]*(b +
Sqrt[b^2 - 4*a*c]))))/3

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^n+c*x^(2*n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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