Optimal. Leaf size=148 \[ \frac {\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {p r \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {p r \log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {q r \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {q r \log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h} \]
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Rubi [A] time = 0.12, antiderivative size = 148, normalized size of antiderivative = 1.00, 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 2391
Rule 2393
Rule 2394
Rule 2494
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*}
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Mathematica [A] time = 0.10, size = 163, normalized size = 1.10 \[ \frac {\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \text {Li}_2\left (\frac {h (a+b x)}{a h-b g}\right )-p r \log (a+b x) \log (g+h x)+p r \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+q r \text {Li}_2\left (\frac {h (c+d x)}{c h-d g}\right )-q r \log (c+d x) \log (g+h x)+q r \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h x + g}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{h x +g}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 186, normalized size = 1.26 \[ \frac {{\left (\frac {{\left (\log \left (b x + a\right ) \log \left (\frac {b h x + a h}{b g - a h} + 1\right ) + {\rm Li}_2\left (-\frac {b h x + a h}{b g - a h}\right )\right )} f p}{h} + \frac {{\left (\log \left (d x + c\right ) \log \left (\frac {d h x + c h}{d g - c h} + 1\right ) + {\rm Li}_2\left (-\frac {d h x + c h}{d g - c h}\right )\right )} f q}{h}\right )} r}{f} - \frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (h x + g\right )}{f h} + \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (h x + g\right )}{h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{g+h\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{g + h x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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