Optimal. Leaf size=65 \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]
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Rubi [A] time = 0.0452679, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {372, 266, 44} \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=-\frac{1}{3 a d e^4 (c+d x)^3}-\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}\\ \end{align*}
Mathematica [A] time = 0.0159609, size = 47, normalized size = 0.72 \[ \frac{b \log \left (a+b (c+d x)^3\right )-\frac{a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,ad{e}^{4} \left ( dx+c \right ) ^{3}}}-{\frac{b\ln \left ( dx+c \right ) }{{a}^{2}d{e}^{4}}}+{\frac{b\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{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982409, size = 157, normalized size = 2.42 \begin{align*} -\frac{1}{3 \,{\left (a d^{4} e^{4} x^{3} + 3 \, a c d^{3} e^{4} x^{2} + 3 \, a c^{2} d^{2} e^{4} x + a c^{3} d e^{4}\right )}} + \frac{b \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 e^{4}} - \frac{b \log \left (d x + c\right )}{a^{2} d e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.56151, size = 355, normalized size = 5.46 \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}\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}\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} d^{4} e^{4} x^{3} + 3 \, a^{2} c d^{3} e^{4} x^{2} + 3 \, a^{2} c^{2} d^{2} e^{4} x + a^{2} c^{3} d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.19192, size = 121, normalized size = 1.86 \begin{align*} - \frac{1}{3 a c^{3} d e^{4} + 9 a c^{2} d^{2} e^{4} x + 9 a c d^{3} e^{4} x^{2} + 3 a d^{4} e^{4} x^{3}} - \frac{b \log{\left (\frac{c}{d} + x \right )}}{a^{2} d e^{4}} + \frac{b \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 e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15164, size = 111, normalized size = 1.71 \begin{align*} \frac{b e^{\left (-4\right )} \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{b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{2} d} - \frac{e^{\left (-4\right )}}{3 \,{\left (d x + c\right )}^{3} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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