Optimal. Leaf size=140 \[ -\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1383, 364} \[ -\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1383
Rubi steps
\begin {align*} \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx &=\frac {(2 c) \int \frac {x^2}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x^2}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {3+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {3+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}\\ \end {align*}
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Mathematica [A] time = 0.65, 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.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{c x^{2 \, n} + b x^{n} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b \,x^{n}+c \,x^{2 n}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{a+b\,x^n+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{a + b x^{n} + c x^{2 n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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