Optimal. Leaf size=431 \[ -\frac {1}{4} \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )\right )^2}{b p r}+8 \left (\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}\right )\right )+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q r \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((a+b x)^{p r}\right )}{b}-\frac {q \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2\left ((a+b x)^{p r}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}-\frac {2 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {2 q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r} \]
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Rubi [A] time = 0.49, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {2496, 6742, 2390, 2302, 30, 2433, 2375, 2317, 2374, 6589, 2396, 2394, 2393, 2391, 6686} \[ -\frac {1}{4} \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (8 \left (\frac {q r \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}\right )+\frac {\left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )\right )^2}{b p r}\right )+\frac {2 p q r^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q^2 r^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {q \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2\left ((a+b x)^{p r}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2317
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2433
Rule 2496
Rule 6589
Rule 6686
Rule 6742
Rubi steps
\begin {align*} \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx &=\int \frac {\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )^2}{a+b x} \, dx-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (2 \int \frac {\log \left ((c+d x)^{q r}\right )}{a+b x} \, dx+\int \frac {\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx\right )\\ &=-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}-\frac {(d q r) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b}\right )\right )+\int \left (\frac {\log ^2\left ((a+b x)^{p r}\right )}{a+b x}+\frac {2 \log \left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{a+b x}+\frac {\log ^2\left ((c+d x)^{q r}\right )}{a+b x}\right ) \, dx\\ &=2 \int \frac {\log \left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{a+b x} \, dx-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}-\frac {(q r) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b}\right )\right )+\int \frac {\log ^2\left ((a+b x)^{p r}\right )}{a+b x} \, dx+\int \frac {\log ^2\left ((c+d x)^{q r}\right )}{a+b x} \, dx\\ &=\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {\operatorname {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{x} \, dx,x,a+b x\right )}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (\left (\frac {b c-a d}{b}+\frac {d x}{b}\right )^{q r}\right )}{x} \, dx,x,a+b x\right )}{b}-\frac {(2 d q r) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log \left ((c+d x)^{q r}\right )}{c+d x} \, dx}{b}\\ &=\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )-\frac {(d q) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 p}+\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\log \left ((a+b x)^{p r}\right )\right )}{b p r}-\frac {(2 q r) \operatorname {Subst}\left (\int \frac {\log \left (x^{q r}\right ) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b}\\ &=\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {(2 q r) \operatorname {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b}-\frac {\left (2 q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b}\\ &=\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )-\frac {2 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\left (2 p q r^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 460, normalized size = 1.07 \[ \frac {6 q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log (a+b x)\right )-3 p r \log ^2(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+3 \log (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 q r \log (a+b x) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 q r \log (c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 p q r^2 \text {Li}_3\left (\frac {d (a+b x)}{a d-b c}\right )-6 p q r^2 \log (a+b x) \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+6 p q r^2 \log ^2(a+b x) \log (c+d x)-3 p q r^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-6 p q r^2 \log (a+b x) \log (c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right )-6 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )+3 q^2 r^2 \log (a+b x) \log ^2(c+d x)-3 q^2 r^2 \log ^2(c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right )+p^2 r^2 \log ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{b x + a}, x\right ) \]
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 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}}{b x +a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\log \left (b x + a\right ) \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right )^{2}}{b} + \int \frac {{\left (r^{2} \log \relax (f)^{2} + 2 \, r \log \relax (e) \log \relax (f) + \log \relax (e)^{2}\right )} b d x + {\left (r^{2} \log \relax (f)^{2} + 2 \, r \log \relax (e) \log \relax (f) + \log \relax (e)^{2}\right )} b c + {\left (b d x + b c\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right )^{2} + 2 \, {\left ({\left (r \log \relax (f) + \log \relax (e)\right )} b d x + {\left (r \log \relax (f) + \log \relax (e)\right )} b c\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right ) + 2 \, {\left ({\left (r \log \relax (f) + \log \relax (e)\right )} b d x + {\left (r \log \relax (f) + \log \relax (e)\right )} b c - {\left (b d q r x + a d q r\right )} \log \left (b x + a\right ) + {\left (b d x + b c\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right )\right )} \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right )}{b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{a+b\,x} \,d x \]
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 \[ \int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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