Optimal. Leaf size=196 \[ \frac{(f x)^{m+1} \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(f x)^{m+1} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (\sqrt{b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.288965, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {1560, 364} \[ \frac{(f x)^{m+1} \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(f x)^{m+1} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (\sqrt{b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
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Rule 1560
Rule 364
Rubi steps
\begin{align*} \int \frac{(f x)^m \left (d+e x^n\right )}{a+b x^n+c x^{2 n}} \, dx &=\int \left (\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^m}{b-\sqrt{b^2-4 a c}+2 c x^n}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^m}{b+\sqrt{b^2-4 a c}+2 c x^n}\right ) \, dx\\ &=\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f x)^m}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx+\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f x)^m}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx\\ &=\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (b-\sqrt{b^2-4 a c}\right ) f (1+m)}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.308321, size = 158, normalized size = 0.81 \[ \frac{x (f x)^m \left (\left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )\right )}{2 a (m+1) \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) }{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{{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{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{{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{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 )} \left (f x\right )^{m}}{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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