Optimal. Leaf size=89 \[ -\frac {3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}+\frac {d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac {(3 c d-b e) \log (d+e x)}{e^4}+\frac {c x}{e^3} \]
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Rubi [A] time = 0.08, antiderivative size = 89, 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 )}{2 e^4 (d+e x)^2}-\frac {3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac {(3 c d-b e) \log (d+e x)}{e^4}+\frac {c x}{e^3} \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)^3} \, dx &=\int \left (\frac {c}{e^3}-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^3}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^2}+\frac {-3 c d+b e}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {c x}{e^3}+\frac {d \left (c d^2-b d e+a e^2\right )}{2 e^4 (d+e x)^2}-\frac {3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac {(3 c d-b e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.90 \begin {gather*} \frac {\frac {d e (a e-b d)+c d^3}{(d+e x)^2}-\frac {2 \left (e (a e-2 b d)+3 c d^2\right )}{d+e x}+2 (b e-3 c d) \log (d+e x)+2 c e x}{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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 147, normalized size = 1.65 \begin {gather*} \frac {2 \, c e^{3} x^{3} + 4 \, c d e^{2} x^{2} - 5 \, c d^{3} + 3 \, b d^{2} e - a d e^{2} - 2 \, {\left (2 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x - 2 \, {\left (3 \, c d^{3} - b d^{2} e + {\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 2 \, {\left (3 \, c d^{2} e - b d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 82, normalized size = 0.92 \begin {gather*} c x e^{\left (-3\right )} - {\left (3 \, c d - b e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \, {\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x\right )} e^{\left (-4\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 = 121, normalized size = 1.36 \begin {gather*} \frac {a d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {b \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {c \,d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {a}{\left (e x +d \right ) e^{2}}+\frac {2 b d}{\left (e x +d \right ) e^{3}}+\frac {b \ln \left (e x +d \right )}{e^{3}}-\frac {3 c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 c d \ln \left (e x +d \right )}{e^{4}}+\frac {c x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 96, normalized size = 1.08 \begin {gather*} -\frac {5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \, {\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac {c x}{e^{3}} - \frac {{\left (3 \, c d - b e\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 = 2.35, size = 96, normalized size = 1.08 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (b\,e-3\,c\,d\right )}{e^4}-\frac {x\,\left (3\,c\,d^2-2\,b\,d\,e+a\,e^2\right )+\frac {5\,c\,d^3-3\,b\,d^2\,e+a\,d\,e^2}{2\,e}}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2}+\frac {c\,x}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 97, normalized size = 1.09 \begin {gather*} \frac {c x}{e^{3}} + \frac {- a d e^{2} + 3 b d^{2} e - 5 c d^{3} + x \left (- 2 a e^{3} + 4 b d e^{2} - 6 c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac {\left (b e - 3 c d\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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