Optimal. Leaf size=107 \[ \frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {q r \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {p r \log ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2494, 2390, 2301, 2394, 2393, 2391} \[ -\frac {q r \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {p r \log ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2390
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 )}{a+b x} \, dx &=\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-(p r) \int \frac {\log (a+b x)}{a+b x} \, dx-\frac {(d q r) \int \frac {\log (a+b x)}{c+d x} \, dx}{b}\\ &=-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {(p r) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b}+(q r) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx\\ &=-\frac {p r \log ^2(a+b x)}{2 b}-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac {(q r) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b}\\ &=-\frac {p r \log ^2(a+b x)}{2 b}-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {q r \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 93, normalized size = 0.87 \[ -\frac {\log (a+b x) \left (-2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+p r \log (a+b x)\right )+2 q r \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{b x + a}, x\right ) \]
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 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{b x +a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 164, normalized size = 1.53 \[ -\frac {{\left (\frac {2 \, {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} f q}{b} - \frac {f p \log \left (b x + a\right )^{2} + 2 \, f q \log \left (b x + a\right ) \log \left (d x + c\right )}{b}\right )} r}{2 \, f} - \frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (b x + a\right )}{b f} + \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (b x + a\right )}{b} \]
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 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{a+b\,x} \,d x \]
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 \[ \int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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