Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]
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Rubi [A] time = 0.0213417, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 260} \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 260
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{a+b (c+d x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\log \left (a+b (c+d x)^3\right )}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.0065988, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 43, normalized size = 2. \begin{align*}{\frac{\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 = 1.16176, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.4815, size = 92, normalized size = 4.18 \begin{align*} \frac{\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 = 0.410921, size = 42, normalized size = 1.91 \begin{align*} \frac{\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 = 1.12219, size = 28, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left |{\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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