Optimal. Leaf size=25 \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]
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Rubi [A] time = 0.0201235, antiderivative size = 25, 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, 260} \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 260
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^2}{a+b (c+d x)^3} \, dx &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.008426, size = 25, normalized size = 1. \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972769, size = 61, normalized size = 2.44 \begin{align*} \frac{e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56174, size = 97, normalized size = 3.88 \begin{align*} \frac{e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.447674, size = 46, normalized size = 1.84 \begin{align*} \frac{e^{2} \log{\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1769, size = 61, normalized size = 2.44 \begin{align*} \frac{e^{2} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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