Optimal. Leaf size=545 \[ -\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c f x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.351151, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1793, 1893, 245, 364} \[ -\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c f x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 1793
Rule 1893
Rule 245
Rule 364
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx &=\frac{(2 c) \int \frac{d+e x+f x^2+g x^3}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{d+e x+f x^2+g x^3}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{(2 c) \int \left (-\frac{d}{-b+\sqrt{b^2-4 a c}-2 c x^n}-\frac{e x}{-b+\sqrt{b^2-4 a c}-2 c x^n}-\frac{f x^2}{-b+\sqrt{b^2-4 a c}-2 c x^n}-\frac{g x^3}{-b+\sqrt{b^2-4 a c}-2 c x^n}\right ) \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \left (\frac{d}{b+\sqrt{b^2-4 a c}+2 c x^n}+\frac{e x}{b+\sqrt{b^2-4 a c}+2 c x^n}+\frac{f x^2}{b+\sqrt{b^2-4 a c}+2 c x^n}+\frac{g x^3}{b+\sqrt{b^2-4 a c}+2 c x^n}\right ) \, dx}{\sqrt{b^2-4 a c}}\\ &=-\frac{(2 c d) \int \frac{1}{-b+\sqrt{b^2-4 a c}-2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c d) \int \frac{1}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c e) \int \frac{x}{-b+\sqrt{b^2-4 a c}-2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c e) \int \frac{x}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c f) \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 f) \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 g) \int \frac{x^3}{-b+\sqrt{b^2-4 a c}-2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c g) \int \frac{x^3}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}\\ &=-\frac{2 c d 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 )}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{2 c d 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 )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{2 c f 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 f 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{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{4+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{c g x^4 \, _2F_1\left (1,\frac{4}{n};\frac{4+n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}\\ \end{align*}
Mathematica [A] time = 0.566578, size = 460, normalized size = 0.84 \[ \frac{x \left (12 d \left (\sqrt{b^2-4 a c}+b\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+12 d \left (\sqrt{b^2-4 a c}-b\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )+x \left (6 e \left (\sqrt{b^2-4 a c}+b\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+6 e \left (\sqrt{b^2-4 a c}-b\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )+x \left (4 f \left (\sqrt{b^2-4 a c}+b\right ) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+4 f \left (\sqrt{b^2-4 a c}-b\right ) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )+3 g x \left (\left (\sqrt{b^2-4 a c}+b\right ) \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (\sqrt{b^2-4 a c}-b\right ) \, _2F_1\left (1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )\right )\right )\right )\right )}{24 a \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{g{x}^{3}+f{x}^{2}+ex+d}{a+b{x}^{n}+c{x}^{2\,n}}}\, 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*} \int \frac{g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a}\,{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{g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a}, 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{g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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