Optimal. Leaf size=148 \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac{p r \log (g+h x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h}-\frac{q r \log (g+h x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h} \]
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Rubi [A] time = 0.120007, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2494, 2394, 2393, 2391} \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h}+\frac{\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac{p r \log (g+h x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h}-\frac{q r \log (g+h x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h} \]
Antiderivative was successfully verified.
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Rule 2494
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx &=\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac{(b p r) \int \frac{\log (g+h x)}{a+b x} \, dx}{h}-\frac{(d q r) \int \frac{\log (g+h x)}{c+d x} \, dx}{h}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+(p r) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx+(q r) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac{(p r) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}+\frac{(q r) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac{p r \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}-\frac{q r \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}\\ \end{align*}
Mathematica [A] time = 0.0834048, size = 163, normalized size = 1.1 \[ \frac{p r \text{PolyLog}\left (2,\frac{h (a+b x)}{a h-b g}\right )+q r \text{PolyLog}\left (2,\frac{h (c+d x)}{c h-d g}\right )+\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log (a+b x) \log (g+h x)+p r \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )-q r \log (c+d x) \log (g+h x)+q r \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{hx+g}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21998, size = 251, normalized size = 1.7 \begin{align*} \frac{{\left (\frac{{\left (\log \left (b x + a\right ) \log \left (\frac{b h x + a h}{b g - a h} + 1\right ) +{\rm Li}_2\left (-\frac{b h x + a h}{b g - a h}\right )\right )} f p}{h} + \frac{{\left (\log \left (d x + c\right ) \log \left (\frac{d h x + c h}{d g - c h} + 1\right ) +{\rm Li}_2\left (-\frac{d h x + c h}{d g - c h}\right )\right )} f q}{h}\right )} r}{f} - \frac{{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (h x + g\right )}{f h} + \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (h x + g\right )}{h} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h x + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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