Optimal. Leaf size=20 \[ \frac{(a e+c d x)^3}{3 c d} \]
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Rubi [A] time = 0.012909, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 32} \[ \frac{(a e+c d x)^3}{3 c d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx &=\int (a e+c d x)^2 \, dx\\ &=\frac{(a e+c d x)^3}{3 c d}\\ \end{align*}
Mathematica [A] time = 0.0019549, size = 20, normalized size = 1. \[ \frac{(a e+c d x)^3}{3 c d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( cdx+ae \right ) ^{3}}{3\,cd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988521, size = 38, normalized size = 1.9 \begin{align*} \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51214, size = 58, normalized size = 2.9 \begin{align*} \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.14883, size = 29, normalized size = 1.45 \begin{align*} a^{2} e^{2} x + a c d e x^{2} + \frac{c^{2} d^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19405, size = 116, normalized size = 5.8 \begin{align*} \frac{1}{3} \,{\left (c^{2} d^{2} - \frac{3 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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