Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]
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Rubi [A] time = 0.0242018, 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} \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 260
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{a+b (c+d x)^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a+b x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{\log \left (a+b (c+d x)^4\right )}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.0087351, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 55, normalized size = 2.5 \begin{align*}{\frac{\ln \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 ) }{4\,bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973005, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.22035, size = 116, normalized size = 5.27 \begin{align*} \frac{\log \left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.467771, size = 56, normalized size = 2.55 \begin{align*} \frac{\log{\left (a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} \right )}}{4 b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19299, size = 27, normalized size = 1.23 \begin{align*} \frac{\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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