Optimal. Leaf size=308 \[ \frac{x \left (\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)} \]
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Rubi [A] time = 0.69909, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1424, 1422, 245} \[ \frac{x \left (\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)} \]
Antiderivative was successfully verified.
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Rule 1424
Rule 1422
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^3}{a+b x^n+c x^{2 n}} \, dx &=\int \left (\frac{e^2 (3 c d-b e)}{c^2}+\frac{e^3 x^n}{c}+\frac{c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{c^2 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^{1+n}}{c (1+n)}+\frac{\int \frac{c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{c^2}\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^{1+n}}{c (1+n)}+\frac{\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\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}{2 c^2}+\frac{\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\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}{2 c^2}\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^{1+n}}{c (1+n)}+\frac{\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\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 )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\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 )}{c^2 \left (b+\sqrt{b^2-4 a c}\right )}\\ \end{align*}
Mathematica [A] time = 0.87564, size = 295, normalized size = 0.96 \[ \frac{x \left (\frac{\left (\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{b-\sqrt{b^2-4 a c}}+\frac{\left (\frac{(b e-2 c d) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}+b}+e^2 (3 c d-b e)+\frac{c e^3 x^n}{n+1}\right )}{c^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{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*} \frac{c e^{3} x x^{n} +{\left (3 \, c d e^{2}{\left (n + 1\right )} - b e^{3}{\left (n + 1\right )}\right )} x}{c^{2}{\left (n + 1\right )}} - \int -\frac{c^{2} d^{3} -{\left (3 \, c d e^{2} - b e^{3}\right )} a +{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} x^{n}}{c^{3} x^{2 \, n} + b c^{2} x^{n} + a c^{2}}\,{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^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{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{{\left (e x^{n} + d\right )}^{3}}{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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