Optimal. Leaf size=15 \[ \frac{c (d+e x)^4}{4 e} \]
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Rubi [A] time = 0.0040956, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 12, 32} \[ \frac{c (d+e x)^4}{4 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^3 \, dx\\ &=c \int (d+e x)^3 \, dx\\ &=\frac{c (d+e x)^4}{4 e}\\ \end{align*}
Mathematica [A] time = 0.0016316, size = 15, normalized size = 1. \[ \frac{c (d+e x)^4}{4 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 36, normalized size = 2.4 \begin{align*}{\frac{c{e}^{3}{x}^{4}}{4}}+dc{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}ec{x}^{2}}{2}}+c{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1008, size = 41, normalized size = 2.73 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}}{4 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.56161, size = 77, normalized size = 5.13 \begin{align*} \frac{1}{4} x^{4} e^{3} c + x^{3} e^{2} d c + \frac{3}{2} x^{2} e d^{2} c + x d^{3} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.075352, size = 39, normalized size = 2.6 \begin{align*} c d^{3} x + \frac{3 c d^{2} e x^{2}}{2} + c d e^{2} x^{3} + \frac{c e^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18386, size = 46, normalized size = 3.07 \begin{align*} \frac{1}{4} \, c x^{4} e^{3} + c d x^{3} e^{2} + \frac{3}{2} \, c d^{2} x^{2} e + c d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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