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 = 0.48, antiderivative size = 296, normalized size of antiderivative = 1.00, 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 65
Rule 68
Rule 830
Rule 960
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*}
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Mathematica [A] time = 0.43, 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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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n}}{c x^{4} + b x^{3} + a x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{n}}{\left (c \,x^{2}+b x +a \right ) x^{2}}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{n}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^n}{x^2\,\left (c\,x^2+b\,x+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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