Optimal. Leaf size=142 \[ \frac {2 c \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x}+\frac {2 c \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x} \]
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Rubi [A]
time = 0.03, antiderivative size = 142, 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 = {1397, 371}
\begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 1397
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )} \, dx &=\frac {(2 c) \int \frac {1}{x^2 \left (b-\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {1}{x^2 \left (b+\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}\\ &=\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 )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x}+\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 )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 240, normalized size = 1.69 \begin {gather*} \frac {2^{1+\frac {1}{n}} c \left (\frac {\left (\frac {c x^n}{b-\sqrt {b^2-4 a c}+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}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )}{-b+\sqrt {b^2-4 a c}-2 c x^n}+\frac {x^{-n} \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{1+\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}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{c}\right )}{\sqrt {b^2-4 a c} (1+n) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{2} \left (a + b x^{n} + c x^{2 n}\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^2\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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