Integrand size = 20, antiderivative size = 140 \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1397, 371} \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
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Rule 371
Rule 1397
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.55 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.89 \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2}{3} c x^3 \left (\frac {1-\left (\frac {x^n}{-\frac {-b+\sqrt {b^2-4 a c}}{2 c}+x^n}\right )^{-3/n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{n},-\frac {3}{n},\frac {-3+n}{n},\frac {b-\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {1-8^{-1/n} \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-3/n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{n},-\frac {3}{n},\frac {-3+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )}\right ) \]
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\[\int \frac {x^{2}}{a +b \,x^{n}+c \,x^{2 n}}d x\]
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\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{2}}{a + b x^{n} + c x^{2 n}}\, dx \]
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\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^2}{a+b\,x^n+c\,x^{2\,n}} \,d x \]
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