Optimal. Leaf size=242 \[ \frac {c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {2 c (e+f x)}{2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f}\right )}{a \left (2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}+\frac {c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {2 c (e+f x)}{2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f}\right )}{a \left (2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a e (1+n)} \]
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Rubi [A]
time = 0.28, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {974, 67, 844,
70} \begin {gather*} \frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\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 (n+1) \left (2 c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )}+\frac {c \left (1-\frac {b}{\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 (n+1) \left (2 c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {f x}{e}+1\right )}{a e (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 844
Rule 974
Rubi steps
\begin {align*} \int \frac {(e+f x)^n}{x \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {(e+f x)^n}{a x}+\frac {(-b-c x) (e+f x)^n}{a \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {(e+f x)^n}{x} \, dx}{a}+\frac {\int \frac {(-b-c x) (e+f x)^n}{a+b x+c x^2} \, dx}{a}\\ &=-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a e (1+n)}+\frac {\int \left (\frac {\left (-c-\frac {b c}{\sqrt {b^2-4 a c}}\right ) (e+f x)^n}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (-c+\frac {b c}{\sqrt {b^2-4 a c}}\right ) (e+f x)^n}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a}\\ &=-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a e (1+n)}-\frac {\left (c \left (1-\frac {b}{\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}-\frac {\left (c \left (1+\frac {b}{\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}\\ &=\frac {c \left (1+\frac {b}{\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 \left (2 c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}+\frac {c \left (1-\frac {b}{\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 \left (2 c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) (1+n)}-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a e (1+n)}\\ \end {align*}
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Mathematica [A]
time = 1.17, size = 331, normalized size = 1.37 \begin {gather*} \frac {(e+f x)^n \left (2 \left (1+\frac {e}{f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {e}{f x}\right )+\frac {2^{-n} \left (-\left (\left (b f+\sqrt {\left (b^2-4 a c\right ) f^2}\right ) \left (\frac {c (e+f x)}{b f-\sqrt {\left (b^2-4 a c\right ) f^2}+2 c f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac {2 c e-b f+\sqrt {\left (b^2-4 a c\right ) f^2}}{-b f+\sqrt {\left (b^2-4 a c\right ) f^2}-2 c f x}\right )\right )-\left (-b f+\sqrt {\left (b^2-4 a c\right ) f^2}\right ) \left (\frac {c (e+f x)}{b f+\sqrt {\left (b^2-4 a c\right ) f^2}+2 c f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac {-2 c e+b f+\sqrt {\left (b^2-4 a c\right ) f^2}}{b f+\sqrt {\left (b^2-4 a c\right ) f^2}+2 c f x}\right )\right )}{\sqrt {\left (b^2-4 a c\right ) f^2}}\right )}{2 a n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{n}}{x \left (c \,x^{2}+b x +a \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (e + f x\right )^{n}}{x \left (a + b x + c x^{2}\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^n}{x\,\left (c\,x^2+b\,x+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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