Optimal. Leaf size=27 \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]
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Rubi [A] time = 0.0595308, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2390, 12, 2302, 30} \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]
Antiderivative was successfully verified.
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Rule 2390
Rule 12
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{(a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac{(a+b \log (c (e+f x)))^3}{3 b d f}\\ \end{align*}
Mathematica [A] time = 0.004426, size = 27, normalized size = 1. \[ \frac{(a+b \log (c (e+f x)))^3}{3 b d f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 63, normalized size = 2.3 \begin{align*}{\frac{{a}^{2}\ln \left ( cfx+ce \right ) }{df}}+{\frac{ab \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{df}}+{\frac{{b}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{3}}{3\,df}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16093, size = 173, normalized size = 6.41 \begin{align*} -a b{\left (\frac{2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac{\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + \frac{b^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac{2 \, a b \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac{a^{2} \log \left (d f x + d e\right )}{d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7152, size = 119, normalized size = 4.41 \begin{align*} \frac{b^{2} \log \left (c f x + c e\right )^{3} + 3 \, a b \log \left (c f x + c e\right )^{2} + 3 \, a^{2} \log \left (c f x + c e\right )}{3 \, d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.485953, size = 51, normalized size = 1.89 \begin{align*} \frac{a^{2} \log{\left (d e + d f x \right )}}{d f} + \frac{a b \log{\left (c \left (e + f x\right ) \right )}^{2}}{d f} + \frac{b^{2} \log{\left (c \left (e + f x\right ) \right )}^{3}}{3 d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2072, size = 72, normalized size = 2.67 \begin{align*} \frac{b^{2} \log \left ({\left (f x + e\right )} c\right )^{3} + 3 \, a b \log \left ({\left (f x + e\right )} c\right )^{2} + 3 \, a^{2} \log \left ({\left (f x + e\right )} c\right )}{3 \, d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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