Optimal. Leaf size=201 \[ \frac {2 c x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) (1+m)}-\frac {2 c x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)} \]
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Rubi [A]
time = 0.34, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {925, 140, 138}
\begin {gather*} \frac {2 c x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 925
Rubi steps
\begin {align*} \int \frac {x^m (e+f x)^n}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c x^m (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 x^m (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 {x^m (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 {x^m (e+f x)^n}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 c (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n}\right ) \int \frac {x^m \left (1+\frac {f x}{e}\right )^n}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 c (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n}\right ) \int \frac {x^m \left (1+\frac {f x}{e}\right )^n}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {2 c x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) (1+m)}-\frac {2 c x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}\\ \end {align*}
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Mathematica [F]
time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m (e+f x)^n}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (f x +e \right )^{n}}{c \,x^{2}+b x +a}\, dx\]
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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {x^m\,{\left (e+f\,x\right )}^n}{c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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