Optimal. Leaf size=23 \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.0188829, antiderivative size = 23, 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, 261} \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{1}{3 b d \left (a+b (c+d x)^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0116112, size = 23, normalized size = 1. \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 44, normalized size = 1.9 \begin{align*} -{\frac{1}{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 [A] time = 0.936266, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{3 \,{\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46935, size = 103, normalized size = 4.48 \begin{align*} -\frac{1}{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 = 1.73067, size = 58, normalized size = 2.52 \begin{align*} - \frac{1}{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 = 1.13199, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{3 \,{\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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