Optimal. Leaf size=62 \[ -\frac {a e^2-b d e+c d^2}{2 e^3 (d+e x)^2}+\frac {2 c d-b e}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 62, 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 {a e^2-b d e+c d^2}{2 e^3 (d+e x)^2}+\frac {2 c d-b e}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \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)^3} \, dx &=\int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^3}+\frac {-2 c d+b e}{e^2 (d+e x)^2}+\frac {c}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {c d^2-b d e+a e^2}{2 e^3 (d+e x)^2}+\frac {2 c d-b e}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.92 \begin {gather*} \frac {-e (a e+b d+2 b e x)+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 87, normalized size = 1.40 \begin {gather*} \frac {3 \, c d^{2} - b d e - a e^{2} + 2 \, {\left (2 \, c d e - b e^{2}\right )} x + 2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 60, normalized size = 0.97 \begin {gather*} c e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (2 \, {\left (2 \, c d - b e\right )} x + {\left (3 \, c d^{2} - b d e - a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.34 \begin {gather*} -\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {b}{\left (e x +d \right ) e^{2}}+\frac {2 c d}{\left (e x +d \right ) e^{3}}+\frac {c \ln \left (e x +d \right )}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 71, normalized size = 1.15 \begin {gather*} \frac {3 \, c d^{2} - b d e - a e^{2} + 2 \, {\left (2 \, c d e - b e^{2}\right )} x}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac {c \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 = 0.07, size = 67, normalized size = 1.08 \begin {gather*} \frac {c\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {-3\,c\,d^2+b\,d\,e+a\,e^2}{2\,e^3}+\frac {x\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 68, normalized size = 1.10 \begin {gather*} \frac {c \log {\left (d + e x \right )}}{e^{3}} + \frac {- a e^{2} - b d e + 3 c d^{2} + x \left (- 2 b e^{2} + 4 c d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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