Optimal. Leaf size=23 \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]
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Rubi [A] time = 0.0255701, 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} \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{\left (a+b (c+d x)^4\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x^4\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{1}{4 b d \left (a+b (c+d x)^4\right )}\\ \end{align*}
Mathematica [A] time = 0.0112555, size = 23, normalized size = 1. \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 56, normalized size = 2.4 \begin{align*} -{\frac{1}{4\,bd \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976203, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.23135, size = 130, normalized size = 5.65 \begin{align*} -\frac{1}{4 \,{\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x +{\left (b^{2} c^{4} + a b\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.23872, size = 73, normalized size = 3.17 \begin{align*} - \frac{1}{4 a b d + 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 = 1.16967, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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