Optimal. Leaf size=283 \[ -\frac{c x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
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Rubi [A] time = 0.385304, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1345, 1422, 245} \[ -\frac{c x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
Antiderivative was successfully verified.
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Rule 1345
Rule 1422
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx &=\frac{x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{\int \frac{b^2-2 a c-\left (b^2-4 a c\right ) n+b c (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{\left (c \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt{b^2-4 a c} (1-n)\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (c \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt{b^2-4 a c} (1-n)\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 a \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac{x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt{b^2-4 a c} (1-n)\right ) 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 )}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n}-\frac{c \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt{b^2-4 a c} (1-n)\right ) 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 )}{a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}\\ \end{align*}
Mathematica [B] time = 0.98965, size = 946, normalized size = 3.34 \[ -\frac{x \left (-2 b c^2 \sqrt{b^2-4 a c} n \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right ) x^n+2 b c^2 \sqrt{b^2-4 a c} \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right ) x^n-\left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) (n+1) \left (b c x^n+b^2-2 a c\right )+8 a c^2 \sqrt{b^2-4 a c} n (n+1) \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-2 b^2 c \sqrt{b^2-4 a c} n (n+1) \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-4 a c^2 \sqrt{b^2-4 a c} (n+1) \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )+2 b^2 c \sqrt{b^2-4 a c} (n+1) \left (\left (c x^n+b\right ) x^n+a\right ) \left (\left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )-\left (b+\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}{a \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) n (n+1) \left (\left (c x^n+b\right ) x^n+a\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b c x x^{n} +{\left (b^{2} - 2 \, a c\right )} x}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}} - \int -\frac{b c{\left (n - 1\right )} x^{n} - 2 \, a c{\left (2 \, n - 1\right )} + b^{2}{\left (n - 1\right )}}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{c^{2} x^{4 \, n} + b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 2 \,{\left (b c x^{n} + a c\right )} x^{2 \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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