Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^4}{4 e} \]
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Rubi [A] time = 0.0047706, 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} \[ \frac{c^2 (d+e x)^4}{4 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{d+e x} \, dx &=\int c^2 (d+e x)^3 \, dx\\ &=c^2 \int (d+e x)^3 \, dx\\ &=\frac{c^2 (d+e x)^4}{4 e}\\ \end{align*}
Mathematica [A] time = 0.0014003, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^4}{4 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 36, normalized size = 2.1 \begin{align*}{c}^{2} \left ({\frac{{e}^{3}{x}^{4}}{4}}+d{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}e{x}^{2}}{2}}+{d}^{3}x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20283, size = 58, normalized size = 3.41 \begin{align*} \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9784, size = 88, normalized size = 5.18 \begin{align*} \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.104091, size = 46, normalized size = 2.71 \begin{align*} c^{2} d^{3} x + \frac{3 c^{2} d^{2} e x^{2}}{2} + c^{2} d e^{2} x^{3} + \frac{c^{2} e^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17281, size = 66, normalized size = 3.88 \begin{align*} \frac{1}{4} \,{\left (c^{2} x^{4} e^{7} + 4 \, c^{2} d x^{3} e^{6} + 6 \, c^{2} d^{2} x^{2} e^{5} + 4 \, c^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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