Optimal. Leaf size=15 \[ \frac{c (d+e x)^5}{5 e} \]
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Rubi [A] time = 0.0044967, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ \frac{c (d+e x)^5}{5 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^4 \, dx\\ &=c \int (d+e x)^4 \, dx\\ &=\frac{c (d+e x)^5}{5 e}\\ \end{align*}
Mathematica [A] time = 0.0029526, size = 15, normalized size = 1. \[ \frac{c (d+e x)^5}{5 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 48, normalized size = 3.2 \begin{align*}{\frac{{e}^{4}c{x}^{5}}{5}}+d{e}^{3}c{x}^{4}+2\,{d}^{2}c{e}^{2}{x}^{3}+2\,c{d}^{3}e{x}^{2}+c{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09967, size = 63, normalized size = 4.2 \begin{align*} \frac{1}{5} \, c e^{4} x^{5} + c d e^{3} x^{4} + 2 \, c d^{2} e^{2} x^{3} + 2 \, c d^{3} e x^{2} + c d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61673, size = 99, normalized size = 6.6 \begin{align*} \frac{1}{5} x^{5} e^{4} c + x^{4} e^{3} d c + 2 x^{3} e^{2} d^{2} c + 2 x^{2} e d^{3} c + x d^{4} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.072436, size = 51, normalized size = 3.4 \begin{align*} c d^{4} x + 2 c d^{3} e x^{2} + 2 c d^{2} e^{2} x^{3} + c d e^{3} x^{4} + \frac{c e^{4} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1605, size = 61, normalized size = 4.07 \begin{align*} \frac{1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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