Optimal. Leaf size=52 \[ \frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {x (c d-b e)}{e^2}+\frac {c x^2}{2 e} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} \frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {x (c d-b e)}{e^2}+\frac {c x^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{d+e x} \, dx &=\int \left (\frac {-c d+b e}{e^2}+\frac {c x}{e}+\frac {c d^2-b d e+a e^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^2}{2 e}+\frac {\left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.92 \begin {gather*} \frac {2 \log (d+e x) \left (e (a e-b d)+c d^2\right )+e x (2 b e-2 c d+c e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 52, normalized size = 1.00 \begin {gather*} \frac {c e^{2} x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 0.98 \begin {gather*} {\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c x^{2} e - 2 \, c d x + 2 \, b x e\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 1.21 \begin {gather*} \frac {c \,x^{2}}{2 e}+\frac {a \ln \left (e x +d \right )}{e}-\frac {b d \ln \left (e x +d \right )}{e^{2}}+\frac {b x}{e}+\frac {c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {c d x}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 50, normalized size = 0.96 \begin {gather*} \frac {c e x^{2} - 2 \, {\left (c d - b e\right )} x}{2 \, e^{2}} + \frac {{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 51, normalized size = 0.98 \begin {gather*} x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^2}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 44, normalized size = 0.85 \begin {gather*} \frac {c x^{2}}{2 e} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) + \frac {\left (a e^{2} - b d e + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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