Optimal. Leaf size=26 \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.0218556, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {372, 261} \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0126833, size = 26, normalized size = 1. \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 47, normalized size = 1.8 \begin{align*} -{\frac{{e}^{2}}{3\,bd \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973787, size = 74, normalized size = 2.85 \begin{align*} -\frac{e^{2}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55946, size = 108, normalized size = 4.15 \begin{align*} -\frac{e^{2}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.76155, size = 60, normalized size = 2.31 \begin{align*} - \frac{e^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15025, size = 61, normalized size = 2.35 \begin{align*} -\frac{e^{2}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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