Optimal. Leaf size=39 \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]
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Rubi [A] time = 0.0278199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {24, 43} \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin{align*} \int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^5} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^5}+\frac{c d e}{(d+e x)^4}\right ) \, dx}{e^2}\\ &=-\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.012282, size = 30, normalized size = 0.77 \[ -\frac{3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 40, normalized size = 1. \begin{align*} -{\frac{a{e}^{2}-c{d}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{cd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17402, size = 89, normalized size = 2.28 \begin{align*} -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44876, size = 136, normalized size = 3.49 \begin{align*} -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.842324, size = 70, normalized size = 1.79 \begin{align*} - \frac{3 a e^{2} + c d^{2} + 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22475, size = 65, normalized size = 1.67 \begin{align*} -\frac{{\left (4 \, c d x^{2} e^{2} + 5 \, c d^{2} x e + c d^{3} + 3 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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