Optimal. Leaf size=102 \[ -\frac{(a+b \log (c (d+e x))) (g \log (c (d+e x))+f)}{e (d+e x)}-\frac{g (a+b \log (c (d+e x))+b)}{e (d+e x)}-\frac{b (g \log (c (d+e x))+f)}{e (d+e x)}-\frac{b g}{e (d+e x)} \]
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Rubi [A] time = 0.110858, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2369, 12, 2304, 2366} \[ -\frac{(a+b \log (c (d+e x))) (g \log (c (d+e x))+f)}{e (d+e x)}-\frac{g (a+b \log (c (d+e x))+b)}{e (d+e x)}-\frac{b (g \log (c (d+e x))+f)}{e (d+e x)}-\frac{b g}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 2369
Rule 12
Rule 2304
Rule 2366
Rubi steps
\begin{align*} \int \frac{(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c^2 (a+b \log (x)) (f+g \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{c e}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{(a+b \log (x)) (f+g \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{e}\\ &=-\frac{b (f+g \log (c (d+e x)))}{e (d+e x)}-\frac{(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{e (d+e x)}-\frac{(c g) \operatorname{Subst}\left (\int \frac{-a \left (1+\frac{b}{a}\right )-b \log (x)}{x^2} \, dx,x,c (d+e x)\right )}{e}\\ &=-\frac{b g}{e (d+e x)}-\frac{g (a+b+b \log (c (d+e x)))}{e (d+e x)}-\frac{b (f+g \log (c (d+e x)))}{e (d+e x)}-\frac{(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.103999, size = 58, normalized size = 0.57 \[ -\frac{(a g+b (f+2 g)) \log (c (d+e x))+a (f+g)+b g \log ^2(c (d+e x))+b (f+2 g)}{e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 184, normalized size = 1.8 \begin{align*} -{\frac{acf}{e \left ( cex+cd \right ) }}-{\frac{acg\ln \left ( cex+cd \right ) }{e \left ( cex+cd \right ) }}-{\frac{acg}{e \left ( cex+cd \right ) }}-{\frac{bcf\ln \left ( cex+cd \right ) }{e \left ( cex+cd \right ) }}-{\frac{bcf}{e \left ( cex+cd \right ) }}-{\frac{bcg \left ( \ln \left ( cex+cd \right ) \right ) ^{2}}{e \left ( cex+cd \right ) }}-2\,{\frac{bcg\ln \left ( cex+cd \right ) }{e \left ( cex+cd \right ) }}-2\,{\frac{bcg}{e \left ( cex+cd \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08332, size = 215, normalized size = 2.11 \begin{align*} -b{\left (\frac{c e}{c e^{3} x + c d e^{2}} + \frac{\log \left (c e x + c d\right )}{e^{2} x + d e}\right )} f - a{\left (\frac{c e}{c e^{3} x + c d e^{2}} + \frac{\log \left (c e x + c d\right )}{e^{2} x + d e}\right )} g - \frac{a f}{e^{2} x + d e} - \frac{{\left (c^{2} \log \left (c e x + c d\right )^{2} + 2 \, c^{2} \log \left (c e x + c d\right ) + 2 \, c^{2}\right )} b g}{{\left (c e x + c d\right )} c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29366, size = 143, normalized size = 1.4 \begin{align*} -\frac{b g \log \left (c e x + c d\right )^{2} +{\left (a + b\right )} f +{\left (a + 2 \, b\right )} g +{\left (b f +{\left (a + 2 \, b\right )} g\right )} \log \left (c e x + c d\right )}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.448629, size = 75, normalized size = 0.74 \begin{align*} - \frac{b g \log{\left (c \left (d + e x\right ) \right )}^{2}}{d e + e^{2} x} + \frac{\left (- a g - b f - 2 b g\right ) \log{\left (c \left (d + e x\right ) \right )}}{d e + e^{2} x} - \frac{a f + a g + b f + 2 b g}{d e + e^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2507, size = 104, normalized size = 1.02 \begin{align*} -\frac{b g e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c e\right )^{2}}{x e + d} - \frac{{\left (b f + a g\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c e\right )}{x e + d} - \frac{{\left (a f + b g\right )} e^{\left (-1\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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