Optimal. Leaf size=15 \[ -\frac{d}{4 b^2 (c+d x)^4} \]
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Rubi [A] time = 0.0032222, antiderivative size = 15, 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} \[ -\frac{d}{4 b^2 (c+d x)^4} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{b c}{d}+b x\right )^2 (c+d x)^3} \, dx &=\frac{d^2 \int \frac{1}{(c+d x)^5} \, dx}{b^2}\\ &=-\frac{d}{4 b^2 (c+d x)^4}\\ \end{align*}
Mathematica [A] time = 0.0053454, size = 15, normalized size = 1. \[ -\frac{d}{4 b^2 (c+d x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 14, normalized size = 0.9 \begin{align*} -{\frac{d}{4\,{b}^{2} \left ( dx+c \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00136, size = 80, normalized size = 5.33 \begin{align*} -\frac{d}{4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44963, size = 116, normalized size = 7.73 \begin{align*} -\frac{d}{4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.48291, size = 68, normalized size = 4.53 \begin{align*} - \frac{d^{2}}{4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06731, size = 27, normalized size = 1.8 \begin{align*} -\frac{b^{2}}{4 \,{\left (b x + \frac{b c}{d}\right )}^{4} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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