Optimal. Leaf size=237 \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{(e+f x)^{n+1}}{c f (n+1)} \]
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Rubi [A] time = 0.382054, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1628, 68} \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{(e+f x)^{n+1}}{c f (n+1)} \]
Antiderivative was successfully verified.
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Rule 1628
Rule 68
Rubi steps
\begin{align*} \int \frac{x^2 (e+f x)^n}{a+b x+c x^2} \, dx &=\int \left (\frac{(e+f x)^n}{c}+\frac{\left (-\frac{b}{c}+\frac{b^2-2 a c}{c \sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (-\frac{b}{c}-\frac{b^2-2 a c}{c \sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx\\ &=\frac{(e+f x)^{1+n}}{c f (1+n)}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{(e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{c}-\frac{\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{(e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{c}\\ &=\frac{(e+f x)^{1+n}}{c f (1+n)}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{c \left (2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}+\frac{\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{c \left (2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.36375, size = 202, normalized size = 0.85 \[ \frac{(e+f x)^{n+1} \left (\frac{\left (\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left (\sqrt{b^2-4 a c}-b\right ) f}\right )}{f \left (\sqrt{b^2-4 a c}-b\right )+2 c e}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{2 c e-f \left (\sqrt{b^2-4 a c}+b\right )}+\frac{1}{f}\right )}{c (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.309, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}{x}^{2}}{c{x}^{2}+bx+a}}\, 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 (f x + e\right )}^{n} x^{2}}{c x^{2} + b x + 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 (f x + e\right )}^{n} x^{2}}{c x^{2} + b x + 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 (f x + e\right )}^{n} x^{2}}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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