Optimal. Leaf size=17 \[ -\frac{1}{4 c e (d+e x)^4} \]
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Rubi [A] time = 0.004754, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{1}{4 c e (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx &=\int \frac{1}{c (d+e x)^5} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^5} \, dx}{c}\\ &=-\frac{1}{4 c e (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.004563, size = 17, normalized size = 1. \[ -\frac{1}{4 c e (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,ce \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00505, size = 69, normalized size = 4.06 \begin{align*} -\frac{1}{4 \,{\left (c e^{5} x^{4} + 4 \, c d e^{4} x^{3} + 6 \, c d^{2} e^{3} x^{2} + 4 \, c d^{3} e^{2} x + c d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85085, size = 105, normalized size = 6.18 \begin{align*} -\frac{1}{4 \,{\left (c e^{5} x^{4} + 4 \, c d e^{4} x^{3} + 6 \, c d^{2} e^{3} x^{2} + 4 \, c d^{3} e^{2} x + c d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.65278, size = 58, normalized size = 3.41 \begin{align*} - \frac{1}{4 c d^{4} e + 16 c d^{3} e^{2} x + 24 c d^{2} e^{3} x^{2} + 16 c d e^{4} x^{3} + 4 c e^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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