Optimal. Leaf size=296 \[ -\frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{a^2 (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{a^2 (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{b (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a^2 e (n+1)}+\frac{f (e+f x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{f x}{e}+1\right )}{a e^2 (n+1)} \]
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Rubi [A] time = 0.477245, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {960, 65, 830, 68} \[ -\frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{a^2 (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{a^2 (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{b (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a^2 e (n+1)}+\frac{f (e+f x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{f x}{e}+1\right )}{a e^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 960
Rule 65
Rule 830
Rule 68
Rubi steps
\begin{align*} \int \frac{(e+f x)^n}{x^2 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{(e+f x)^n}{a x^2}-\frac{b (e+f x)^n}{a^2 x}+\frac{\left (b^2-a c+b c x\right ) (e+f x)^n}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (b^2-a c+b c x\right ) (e+f x)^n}{a+b x+c x^2} \, dx}{a^2}+\frac{\int \frac{(e+f x)^n}{x^2} \, dx}{a}-\frac{b \int \frac{(e+f x)^n}{x} \, dx}{a^2}\\ &=\frac{b (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{f x}{e}\right )}{a^2 e (1+n)}+\frac{f (e+f x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac{f x}{e}\right )}{a e^2 (1+n)}+\frac{\int \left (\frac{\left (b c+\frac{c \left (b^2-2 a c\right )}{\sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (b c-\frac{c \left (b^2-2 a c\right )}{\sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{a^2}\\ &=\frac{b (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{f x}{e}\right )}{a^2 e (1+n)}+\frac{f (e+f x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac{f x}{e}\right )}{a e^2 (1+n)}+\frac{\left (c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{(e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{a^2}+\frac{\left (c \left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{(e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{a^2}\\ &=-\frac{c \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 )}{a^2 \left (2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}-\frac{c \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 )}{a^2 \left (2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}+\frac{b (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{f x}{e}\right )}{a^2 e (1+n)}+\frac{f (e+f x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac{f x}{e}\right )}{a e^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.438454, size = 246, normalized size = 0.83 \[ \frac{(e+f x)^{n+1} \left (-\frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left (\sqrt{b^2-4 a c}-b\right ) f}\right )}{f \left (\sqrt{b^2-4 a c}-b\right )+2 c e}-\frac{c \left (\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{2 c e-f \left (\sqrt{b^2-4 a c}+b\right )}+\frac{a f \, _2F_1\left (2,n+1;n+2;\frac{f x}{e}+1\right )}{e^2}+\frac{b \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{e}\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.307, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}}{{x}^{2} \left ( c{x}^{2}+bx+a \right ) }}\, 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}}{{\left (c x^{2} + b x + a\right )} x^{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{{\left (f x + e\right )}^{n}}{c x^{4} + b x^{3} + a x^{2}}, 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}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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