Optimal. Leaf size=59 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac{\log (c+d x)}{a^2 d}+\frac{1}{3 a d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.0571008, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {372, 266, 44} \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac{\log (c+d x)}{a^2 d}+\frac{1}{3 a d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac{1}{3 a d \left (a+b (c+d x)^3\right )}+\frac{\log (c+d x)}{a^2 d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0223228, size = 48, normalized size = 0.81 \[ \frac{\frac{a}{a+b (c+d x)^3}-\log \left (a+b (c+d x)^3\right )+3 \log (c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 100, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{{a}^{2}d}}+{\frac{1}{3\,ad \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}-{\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\,{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02442, size = 140, normalized size = 2.37 \begin{align*} \frac{1}{3 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )}} - \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^{2} d} + \frac{\log \left (d x + c\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54478, size = 367, normalized size = 6.22 \begin{align*} -\frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \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 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} b d^{4} x^{3} + 3 \, a^{2} b c d^{3} x^{2} + 3 \, a^{2} b c^{2} d^{2} x +{\left (a^{2} b c^{3} + a^{3}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.253, size = 110, normalized size = 1.86 \begin{align*} \frac{1}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac{\log{\left (\frac{c}{d} + x \right )}}{a^{2} 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^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13455, size = 136, normalized size = 2.31 \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^{2} d} + \frac{\log \left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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