Optimal. Leaf size=39 \[ -\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2} \]
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Rubi [A] time = 0.0276398, 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} \[ -\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2} \]
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)^5} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^4} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^4}+\frac{c d e}{(d+e x)^3}\right ) \, dx}{e^2}\\ &=-\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0118837, size = 30, normalized size = 0.77 \[ -\frac{2 a e^2+c d (d+3 e x)}{6 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 40, normalized size = 1. \begin{align*} -{\frac{cd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}-c{d}^{2}}{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 = 1.12517, size = 74, normalized size = 1.9 \begin{align*} -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56125, size = 113, normalized size = 2.9 \begin{align*} -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.85635, size = 58, normalized size = 1.49 \begin{align*} - \frac{2 a e^{2} + c d^{2} + 3 c d e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27447, size = 57, normalized size = 1.46 \begin{align*} -\frac{c d e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} + \frac{c d^{2} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{3}} - \frac{a}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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