Optimal. Leaf size=154 \[ \frac{x \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{b-\sqrt{b^2-4 a c}}+\frac{x \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}+b} \]
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Rubi [A] time = 0.120234, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1422, 245} \[ \frac{x \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{b-\sqrt{b^2-4 a c}}+\frac{x \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}+b} \]
Antiderivative was successfully verified.
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Rule 1422
Rule 245
Rubi steps
\begin{align*} \int \frac{d+e x^n}{a+b x^n+c x^{2 n}} \, dx &=\frac{1}{2} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx+\frac{1}{2} \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx\\ &=\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\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 )}{b-\sqrt{b^2-4 a c}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\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 )}{b+\sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.0754941, size = 134, normalized size = 0.87 \[ \frac{x \left (\left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )\right )}{2 a \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{d+e{x}^{n}}{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{e x^{n} + 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{e x^{n} + 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x^{n}}{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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{n} + 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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