Optimal. Leaf size=69 \[ -\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0464884, 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} \[ -\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2} \]
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)^5} \, dx &=\int \left (\frac{c d^2-b d e+a e^2}{e^2 (d+e x)^5}+\frac{-2 c d+b e}{e^2 (d+e x)^4}+\frac{c}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac{c d^2-b d e+a e^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0182494, size = 49, normalized size = 0.71 \[ -\frac{e (3 a e+b (d+4 e x))+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 63, normalized size = 0.9 \begin{align*} -{\frac{a{e}^{2}-bde+c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{be-2\,cd}{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 = 1.00307, size = 116, normalized size = 1.68 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.899, size = 180, normalized size = 2.61 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49977, size = 92, normalized size = 1.33 \begin{align*} - \frac{3 a e^{2} + b d e + c d^{2} + 6 c e^{2} x^{2} + x \left (4 b e^{2} + 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07926, size = 116, normalized size = 1.68 \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, c e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac{4 \, b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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