Optimal. Leaf size=81 \[ \log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \text {Li}_2\left (-\frac {b x}{a}\right )-p r \log (x) \log \left (\frac {b x}{a}+1\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right )-q r \log (x) \log \left (\frac {d x}{c}+1\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2494, 2317, 2391} \[ -p r \text {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \text {PolyLog}\left (2,-\frac {d x}{c}\right )+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log (x) \log \left (\frac {b x}{a}+1\right )-q r \log (x) \log \left (\frac {d x}{c}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2317
Rule 2391
Rule 2494
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx &=\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-(b p r) \int \frac {\log (x)}{a+b x} \, dx-(d q r) \int \frac {\log (x)}{c+d x} \, dx\\ &=-p r \log (x) \log \left (1+\frac {b x}{a}\right )+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-q r \log (x) \log \left (1+\frac {d x}{c}\right )+(p r) \int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx+(q r) \int \frac {\log \left (1+\frac {d x}{c}\right )}{x} \, dx\\ &=-p r \log (x) \log \left (1+\frac {b x}{a}\right )+\log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-q r \log (x) \log \left (1+\frac {d x}{c}\right )-p r \text {Li}_2\left (-\frac {b x}{a}\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 0.96 \[ \log (x) \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log \left (\frac {b x}{a}+1\right )-q r \log \left (\frac {d x}{c}+1\right )\right )-p r \text {Li}_2\left (-\frac {b x}{a}\right )-q r \text {Li}_2\left (-\frac {d x}{c}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 126, normalized size = 1.56 \[ -\frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \relax (x)}{f} + \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \relax (x) + \frac {{\left ({\left (\log \left (b x + a\right ) \log \left (-\frac {b x + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{a}\right )\right )} f p + {\left (\log \left (d x + c\right ) \log \left (-\frac {d x + c}{c} + 1\right ) + {\rm Li}_2\left (\frac {d x + c}{c}\right )\right )} f q\right )} r}{f} \]
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 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{x} \,d x \]
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 \[ \int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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