Optimal. Leaf size=17 \[ \frac{(d+e x)^4}{4 c^2 e} \]
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Rubi [A] time = 0.0049706, 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} \[ \frac{(d+e x)^4}{4 c^2 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^3}{c^2} \, dx\\ &=\frac{\int (d+e x)^3 \, dx}{c^2}\\ &=\frac{(d+e x)^4}{4 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0010034, size = 17, normalized size = 1. \[ \frac{(d+e x)^4}{4 c^2 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{4}}{4\,{c}^{2}e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02417, size = 50, normalized size = 2.94 \begin{align*} \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06592, size = 77, normalized size = 4.53 \begin{align*} \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.162961, size = 46, normalized size = 2.71 \begin{align*} \frac{d^{3} x}{c^{2}} + \frac{3 d^{2} e x^{2}}{2 c^{2}} + \frac{d e^{2} x^{3}}{c^{2}} + \frac{e^{3} x^{4}}{4 c^{2}} \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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