Optimal. Leaf size=17 \[ -\frac{b^2}{3 d^3 (a+b x)^3} \]
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Rubi [A] time = 0.0033228, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ -\frac{b^2}{3 d^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \left (\frac{a d}{b}+d x\right )^3} \, dx &=\frac{b^3 \int \frac{1}{(a+b x)^4} \, dx}{d^3}\\ &=-\frac{b^2}{3 d^3 (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0048979, size = 17, normalized size = 1. \[ -\frac{b^2}{3 d^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 16, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{3\,{d}^{3} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.99459, size = 63, normalized size = 3.71 \begin{align*} -\frac{b^{2}}{3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64416, size = 92, normalized size = 5.41 \begin{align*} -\frac{b^{2}}{3 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.404026, size = 53, normalized size = 3.12 \begin{align*} - \frac{b^{3}}{3 a^{3} b d^{3} + 9 a^{2} b^{2} d^{3} x + 9 a b^{3} d^{3} x^{2} + 3 b^{4} d^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07017, size = 20, normalized size = 1.18 \begin{align*} -\frac{b^{2}}{3 \,{\left (b x + a\right )}^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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