Optimal. Leaf size=84 \[ \frac {\log (d+e x) \left (3 c d^2-e (2 b d-a e)\right )}{e^4}+\frac {d \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac {x (2 c d-b e)}{e^3}+\frac {c x^2}{2 e^2} \]
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Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {771} \begin {gather*} \frac {d \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}+\frac {\log (d+e x) \left (3 c d^2-e (2 b d-a e)\right )}{e^4}-\frac {x (2 c d-b e)}{e^3}+\frac {c x^2}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac {-2 c d+b e}{e^3}+\frac {c x}{e^2}-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^2}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {(2 c d-b e) x}{e^3}+\frac {c x^2}{2 e^2}+\frac {d \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.94 \begin {gather*} \frac {\frac {2 \left (d e (a e-b d)+c d^3\right )}{d+e x}+2 \log (d+e x) \left (e (a e-2 b d)+3 c d^2\right )+2 e x (b e-2 c d)+c e^2 x^2}{2 e^4} \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 {x \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 131, normalized size = 1.56 \begin {gather*} \frac {c e^{3} x^{3} + 2 \, c d^{3} - 2 \, b d^{2} e + 2 \, a d e^{2} - {\left (3 \, c d e^{2} - 2 \, b e^{3}\right )} x^{2} - 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} x + 2 \, {\left (3 \, c d^{3} - 2 \, b d^{2} e + a d e^{2} + {\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 131, normalized size = 1.56 \begin {gather*} \frac {1}{2} \, {\left ({\left (x e + d\right )}^{2} {\left (c - \frac {2 \, {\left (3 \, c d e - b e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-3\right )} - 2 \, {\left (3 \, c d^{2} - 2 \, b d e + a e^{2}\right )} e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + 2 \, {\left (\frac {c d^{3} e^{2}}{x e + d} - \frac {b d^{2} e^{3}}{x e + d} + \frac {a d e^{4}}{x e + d}\right )} e^{\left (-5\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 108, normalized size = 1.29 \begin {gather*} \frac {c \,x^{2}}{2 e^{2}}+\frac {a d}{\left (e x +d \right ) e^{2}}+\frac {a \ln \left (e x +d \right )}{e^{2}}-\frac {b \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 b d \ln \left (e x +d \right )}{e^{3}}+\frac {b x}{e^{2}}+\frac {c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 c d x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 85, normalized size = 1.01 \begin {gather*} \frac {c d^{3} - b d^{2} e + a d e^{2}}{e^{5} x + d e^{4}} + \frac {c e x^{2} - 2 \, {\left (2 \, c d - b e\right )} x}{2 \, e^{3}} + \frac {{\left (3 \, c d^{2} - 2 \, b d e + a e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 88, normalized size = 1.05 \begin {gather*} x\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )+\frac {c\,x^2}{2\,e^2}+\frac {\ln \left (d+e\,x\right )\,\left (3\,c\,d^2-2\,b\,d\,e+a\,e^2\right )}{e^4}+\frac {c\,d^3-b\,d^2\,e+a\,d\,e^2}{e\,\left (x\,e^4+d\,e^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 82, normalized size = 0.98 \begin {gather*} \frac {c x^{2}}{2 e^{2}} + x \left (\frac {b}{e^{2}} - \frac {2 c d}{e^{3}}\right ) + \frac {a d e^{2} - b d^{2} e + c d^{3}}{d e^{4} + e^{5} x} + \frac {\left (a e^{2} - 2 b d e + 3 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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