Optimal. Leaf size=36 \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]
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Rubi [A] time = 0.0277494, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {372, 266, 36, 29, 31} \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) \left (a+b (c+d x)^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,(c+d x)^3\right )}{3 a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,(c+d x)^3\right )}{3 a d}\\ &=\frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0085122, size = 36, normalized size = 1. \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{ad}}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30615, size = 76, normalized size = 2.11 \begin{align*} -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d} + \frac{\log \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44372, size = 119, normalized size = 3.31 \begin{align*} -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.441534, size = 49, normalized size = 1.36 \begin{align*} \frac{\log{\left (\frac{c}{d} + x \right )}}{a d} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12358, size = 78, normalized size = 2.17 \begin{align*} -\frac{\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac{\log \left ({\left | d x + c \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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