Optimal. Leaf size=191 \[ \frac {2 c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f}\right )}{(n+1) \sqrt {b^2-4 a c} \left (2 c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {2 c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f}\right )}{(n+1) \sqrt {b^2-4 a c} \left (2 c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )} \]
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Rubi [A] time = 0.26, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {711, 68} \[ \frac {2 c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f}\right )}{(n+1) \sqrt {b^2-4 a c} \left (2 c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {2 c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f}\right )}{(n+1) \sqrt {b^2-4 a c} \left (2 c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 711
Rubi steps
\begin {align*} \int \frac {(e+f x)^n}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c (e+f x)^n}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c (e+f x)^n}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {(e+f x)^n}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {(e+f x)^n}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {2 c (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 )}{\sqrt {b^2-4 a c} \left (2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}+\frac {2 c (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 )}{\sqrt {b^2-4 a c} \left (2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 163, normalized size = 0.85 \[ \frac {2 c (e+f x)^{n+1} \left (\frac {\, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f}\right )}{2 c e-f \left (\sqrt {b^2-4 a c}+b\right )}-\frac {\, _2F_1\left (1,n+1;n+2;\frac {2 c (e+f x)}{2 c e+\left (\sqrt {b^2-4 a c}-b\right ) f}\right )}{f \left (\sqrt {b^2-4 a c}-b\right )+2 c e}\right )}{(n+1) \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n}}{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{n}}{c \,x^{2}+b x +a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e+f\,x\right )}^n}{c\,x^2+b\,x+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{n}}{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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