Optimal. Leaf size=69 \[ -\frac{a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.0442678, antiderivative size = 69, normalized size of antiderivative = 1., 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} \[ -\frac{a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3} \]
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)^6} \, dx &=\int \left (\frac{c d^2-b d e+a e^2}{e^2 (d+e x)^6}+\frac{-2 c d+b e}{e^2 (d+e x)^5}+\frac{c}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac{c d^2-b d e+a e^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0204608, size = 51, normalized size = 0.74 \[ -\frac{3 e (4 a e+b (d+5 e x))+2 c \left (d^2+5 d e x+10 e^2 x^2\right )}{60 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 63, normalized size = 0.9 \begin{align*} -{\frac{be-2\,cd}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{a{e}^{2}-bde+c{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{c}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00236, size = 136, normalized size = 1.97 \begin{align*} -\frac{20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \,{\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01564, size = 217, normalized size = 3.14 \begin{align*} -\frac{20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \,{\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.10594, size = 107, normalized size = 1.55 \begin{align*} - \frac{12 a e^{2} + 3 b d e + 2 c d^{2} + 20 c e^{2} x^{2} + x \left (15 b e^{2} + 10 c d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11487, size = 69, normalized size = 1. \begin{align*} -\frac{{\left (20 \, c x^{2} e^{2} + 10 \, c d x e + 2 \, c d^{2} + 15 \, b x e^{2} + 3 \, b d e + 12 \, a e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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