Integrand size = 29, antiderivative size = 540 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q^2 r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(b c-a d)^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d^2} \]
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Time = 0.28 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2584, 2581, 45, 2594, 2579, 31, 8, 2580, 2441, 2440, 2438, 2437, 2338} \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {q r (b c-a d)^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {p q r^2 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d^2}+\frac {p q r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac {p q r^2 (b c-a d)^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b d^2}+\frac {q^2 r^2 (b c-a d)^2 \log ^2(c+d x)}{2 b d^2}+\frac {3 q^2 r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {q r (a+b x) (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p q r^2 x (b c-a d)}{2 d}-\frac {q r^2 x (p+q) (b c-a d)}{d}-\frac {q^2 r^2 x (b c-a d)}{2 d}+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2 \]
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Rule 8
Rule 31
Rule 45
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2579
Rule 2580
Rule 2581
Rule 2584
Rule 2594
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-(p r) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx-\frac {(d q r) \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{b} \\ & = -\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(d q r) \int \left (-\frac {b (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac {(-b c+a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2 (c+d x)}\right ) \, dx}{b}+\frac {1}{2} \left (p^2 r^2\right ) \int (a+b x) \, dx+\frac {\left (d p q r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{2 b} \\ & = \frac {1}{2} a p^2 r^2 x+\frac {1}{4} b p^2 r^2 x^2-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-(q r) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx+\frac {((b c-a d) q r) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{d}-\frac {\left ((b c-a d)^2 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{b d}+\frac {\left (d p q r^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{2 b} \\ & = \frac {1}{2} a p^2 r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}+\frac {1}{4} b p^2 r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {1}{2} \left (p q r^2\right ) \int (a+b x) \, dx+\frac {\left ((b c-a d)^2 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d^2}+\frac {\left (d q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{2 b}+\frac {\left ((b c-a d)^2 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{b d}+\frac {\left ((b c-a d)^2 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b d}-\frac {\left ((b c-a d) q (p+q) r^2\right ) \int 1 \, dx}{d} \\ & = \frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {\left ((b c-a d)^2 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d}+\frac {\left (d q^2 r^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{2 b}+\frac {\left ((b c-a d)^2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b d^2} \\ & = \frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q^2 r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {\left ((b c-a d)^2 p q r^2\right ) \text {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} \\ & = \frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q^2 r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(b c-a d)^2 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b d^2} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.45 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-4 a b c d p q r^2+8 a^2 d^2 p q r^2+2 a b d^2 p^2 r^2 x-6 b^2 c d p q r^2 x+10 a b d^2 p q r^2 x-6 b^2 c d q^2 r^2 x+8 a b d^2 q^2 r^2 x+b^2 d^2 p^2 r^2 x^2+2 b^2 d^2 p q r^2 x^2+b^2 d^2 q^2 r^2 x^2-2 a^2 d^2 p^2 r^2 \log ^2(a+b x)+2 b^2 c^2 p q r^2 \log (c+d x)-4 a b c d p q r^2 \log (c+d x)+8 a^2 d^2 p q r^2 \log (c+d x)+6 b^2 c^2 q^2 r^2 \log (c+d x)-8 a b c d q^2 r^2 \log (c+d x)+2 b^2 c^2 q^2 r^2 \log ^2(c+d x)-4 a b c d q^2 r^2 \log ^2(c+d x)-8 a^2 d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b d^2 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 b^2 c d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-8 a b d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 b^2 c^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+8 a b c d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 a b d^2 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b^2 d^2 x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 p r \log (a+b x) \left (2 b c (b c-2 a d) q r \log (c+d x)-2 (b c-a d)^2 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (3 a d (p-q) r+2 b c q r+2 a d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-4 (b c-a d)^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{4 b d^2} \]
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\[\int \left (b x +a \right ) {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
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\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right ) \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.93 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {{\left (\frac {2 \, a^{2} f p \log \left (b x + a\right )}{b} - \frac {b d f {\left (p + q\right )} x^{2} + 2 \, {\left (a d f {\left (p + 2 \, q\right )} - b c f q\right )} x}{d} - \frac {2 \, {\left (b c^{2} f q - 2 \, a c d f q\right )} \log \left (d x + c\right )}{d^{2}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, f} + \frac {r^{2} {\left (\frac {2 \, {\left ({\left (p q + 3 \, q^{2}\right )} b c^{2} f^{2} - 2 \, {\left (p q + 2 \, q^{2}\right )} a c d f^{2}\right )} \log \left (d x + c\right )}{d^{2}} - \frac {4 \, {\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q + a^{2} d^{2} f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d^{2}} - \frac {2 \, a^{2} d^{2} f^{2} p^{2} \log \left (b x + a\right )^{2} - {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{2} d^{2} f^{2} x^{2} - 4 \, {\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 2 \, {\left (b^{2} c^{2} f^{2} q^{2} - 2 \, a b c d f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (p q + q^{2}\right )} b^{2} c d f^{2} - {\left (p^{2} + 5 \, p q + 4 \, q^{2}\right )} a b d^{2} f^{2}\right )} x - 2 \, {\left (2 \, a b c d f^{2} p q - {\left (p^{2} + 3 \, p q\right )} a^{2} d^{2} f^{2}\right )} \log \left (b x + a\right )}{b d^{2}}\right )}}{4 \, f^{2}} \]
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\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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Timed out. \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,\left (a+b\,x\right ) \,d x \]
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