Optimal. Leaf size=269 \[ -\frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{2 r (p+q) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 q r (b c-a d) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{2 q r^2 (p+q) (b c-a d) \log (c+d x)}{b d}-\frac{2 p q r^2 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d}-\frac{q^2 r^2 (b c-a d) \log ^2(c+d x)}{b d}+2 r^2 x (p+q)^2 \]
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Rubi [A] time = 0.153009, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2487, 2494, 2394, 2393, 2391, 2390, 2301, 31, 8} \[ -\frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{2 r (p+q) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 q r (b c-a d) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{2 q r^2 (p+q) (b c-a d) \log (c+d x)}{b d}-\frac{2 p q r^2 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d}-\frac{q^2 r^2 (b c-a d) \log ^2(c+d x)}{b d}+2 r^2 x (p+q)^2 \]
Antiderivative was successfully verified.
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Rule 2487
Rule 2494
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rule 31
Rule 8
Rubi steps
\begin{align*} \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{(2 (b c-a d) q r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{b}-(2 (p+q) r) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\\ &=-\frac{2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{\left (2 (b c-a d) p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d}-\frac{\left (2 (b c-a d) q^2 r^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b}-\frac{\left (2 (b c-a d) q (p+q) r^2\right ) \int \frac{1}{c+d x} \, dx}{b}+\left (2 (p+q)^2 r^2\right ) \int 1 \, dx\\ &=2 (p+q)^2 r^2 x-\frac{2 (b c-a d) q (p+q) r^2 \log (c+d x)}{b d}-\frac{2 (b c-a d) p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d}-\frac{2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{\left (2 (b c-a d) p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b}-\frac{\left (2 (b c-a d) q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b d}\\ &=2 (p+q)^2 r^2 x-\frac{2 (b c-a d) q (p+q) r^2 \log (c+d x)}{b d}-\frac{2 (b c-a d) p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d}-\frac{(b c-a d) q^2 r^2 \log ^2(c+d x)}{b d}-\frac{2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{\left (2 (b c-a d) p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b d}\\ &=2 (p+q)^2 r^2 x-\frac{2 (b c-a d) q (p+q) r^2 \log (c+d x)}{b d}-\frac{2 (b c-a d) p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d}-\frac{(b c-a d) q^2 r^2 \log ^2(c+d x)}{b d}-\frac{2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{2 (b c-a d) p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b d}\\ \end{align*}
Mathematica [A] time = 0.215741, size = 437, normalized size = 1.62 \[ \frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+b d x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 p r \log (a+b x) \left (d \left (a \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a r (p-q)+b p r x\right )+q r (b c-a d) \log \left (\frac{b (c+d x)}{b c-a d}\right )-b c q r \log (c+d x)\right )-d p^2 r^2 (2 a+b x) \log ^2(a+b x)+2 a d p q r^2 \log (c+d x)+2 a d p q r^2-2 b c p q r^2 \log (c+d x)-b c q^2 r^2 \log ^2(c+d x)-2 b c q^2 r^2 \log (c+d x)+4 b d p q r^2 x+2 b d q^2 r^2 x}{b d}+\frac{p^2 r^2 \left ((a+b x) \log ^2(a+b x)-2 (a+b x) \log (a+b x)+2 b x\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.131, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36595, size = 402, normalized size = 1.49 \begin{align*} x \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} - \frac{2 \,{\left (f{\left (p + q\right )} x - \frac{a f p \log \left (b x + a\right )}{b} - \frac{c f q \log \left (d x + c\right )}{d}\right )} r \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{f} - \frac{{\left (\frac{2 \,{\left (p q + q^{2}\right )} c f^{2} \log \left (d x + c\right )}{d} - \frac{2 \,{\left (b c f^{2} p q - a d f^{2} p q\right )}{\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 )}}{b d} + \frac{a d f^{2} p^{2} \log \left (b x + a\right )^{2} + 2 \, b c f^{2} p q \log \left (b x + a\right ) \log \left (d x + c\right ) + b c f^{2} q^{2} \log \left (d x + c\right )^{2} - 2 \,{\left (p^{2} + 2 \, p q + q^{2}\right )} b d f^{2} x + 2 \,{\left (p^{2} + p q\right )} a d f^{2} \log \left (b x + a\right )}{b d}\right )} r^{2}}{f^{2}} \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 (\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, 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 \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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