Optimal. Leaf size=96 \[ -\frac {3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac {d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c d-b e}{e^4 (d+e x)}+\frac {c \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.08, antiderivative size = 96, 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} \[ -\frac {3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac {d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c d-b e}{e^4 (d+e x)}+\frac {c \log (d+e x)}{e^4} \]
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)^4} \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^4}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^3}+\frac {-3 c d+b e}{e^3 (d+e x)^2}+\frac {c}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^3}-\frac {3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac {3 c d-b e}{e^4 (d+e x)}+\frac {c \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 86, normalized size = 0.90 \[ \frac {-e \left (a e (d+3 e x)+2 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c d \left (11 d^2+27 d e x+18 e^2 x^2\right )+6 c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 141, normalized size = 1.47 \[ \frac {11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x + 6 \, {\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 0.92 \[ c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (6 \, {\left (3 \, c d e - b e^{2}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} - 2 \, b d e - a e^{2}\right )} x + {\left (11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 128, normalized size = 1.33 \[ \frac {a d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {a}{2 \left (e x +d \right )^{2} e^{2}}+\frac {b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 c \,d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {b}{\left (e x +d \right ) e^{3}}+\frac {3 c d}{\left (e x +d \right ) e^{4}}+\frac {c \ln \left (e x +d \right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 113, normalized size = 1.18 \[ \frac {11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {c \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 107, normalized size = 1.11 \[ \frac {c\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {-11\,c\,d^3+2\,b\,d^2\,e+a\,d\,e^2}{6\,e^4}+\frac {x\,\left (-9\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{2\,e^3}+\frac {x^2\,\left (b\,e-3\,c\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.33, size = 114, normalized size = 1.19 \[ \frac {c \log {\left (d + e x \right )}}{e^{4}} + \frac {- a d e^{2} - 2 b d^{2} e + 11 c d^{3} + x^{2} \left (- 6 b e^{3} + 18 c d e^{2}\right ) + x \left (- 3 a e^{3} - 6 b d e^{2} + 27 c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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