Optimal. Leaf size=17 \[ -\frac{1}{7 c^3 e (d+e x)^7} \]
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Rubi [A] time = 0.0047815, 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} \[ -\frac{1}{7 c^3 e (d+e x)^7} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \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)^8} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^8} \, dx}{c^3}\\ &=-\frac{1}{7 c^3 e (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.0045791, size = 17, normalized size = 1. \[ -\frac{1}{7 c^3 e (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,{c}^{3}e \left ( ex+d \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02177, size = 139, normalized size = 8.18 \begin{align*} -\frac{1}{7 \,{\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01563, size = 205, normalized size = 12.06 \begin{align*} -\frac{1}{7 \,{\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.03015, size = 112, normalized size = 6.59 \begin{align*} - \frac{1}{7 c^{3} d^{7} e + 49 c^{3} d^{6} e^{2} x + 147 c^{3} d^{5} e^{3} x^{2} + 245 c^{3} d^{4} e^{4} x^{3} + 245 c^{3} d^{3} e^{5} x^{4} + 147 c^{3} d^{2} e^{6} x^{5} + 49 c^{3} d e^{7} x^{6} + 7 c^{3} e^{8} x^{7}} \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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