Optimal. Leaf size=17 \[ -\frac{1}{5 c^3 e (d+e x)^5} \]
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Rubi [A] time = 0.0040724, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {27, 12, 32} \[ -\frac{1}{5 c^3 e (d+e x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac{1}{c^3 (d+e x)^6} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^6} \, dx}{c^3}\\ &=-\frac{1}{5 c^3 e (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0027721, size = 17, normalized size = 1. \[ -\frac{1}{5 c^3 e (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,{c}^{3}e \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.978386, size = 101, normalized size = 5.94 \begin{align*} -\frac{1}{5 \,{\left (c^{3} e^{6} x^{5} + 5 \, c^{3} d e^{5} x^{4} + 10 \, c^{3} d^{2} e^{4} x^{3} + 10 \, c^{3} d^{3} e^{3} x^{2} + 5 \, c^{3} d^{4} e^{2} x + c^{3} d^{5} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98565, size = 149, normalized size = 8.76 \begin{align*} -\frac{1}{5 \,{\left (c^{3} e^{6} x^{5} + 5 \, c^{3} d e^{5} x^{4} + 10 \, c^{3} d^{2} e^{4} x^{3} + 10 \, c^{3} d^{3} e^{3} x^{2} + 5 \, c^{3} d^{4} e^{2} x + c^{3} d^{5} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.745794, size = 82, normalized size = 4.82 \begin{align*} - \frac{1}{5 c^{3} d^{5} e + 25 c^{3} d^{4} e^{2} x + 50 c^{3} d^{3} e^{3} x^{2} + 50 c^{3} d^{2} e^{4} x^{3} + 25 c^{3} d e^{5} x^{4} + 5 c^{3} e^{6} x^{5}} \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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