Integrand size = 35, antiderivative size = 288 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}-\frac {2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac {2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]
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Time = 0.61 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2465, 2443, 2481, 2421, 6724, 2495} \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\frac {2 b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac {2 b p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac {2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac {2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]
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Rule 2421
Rule 2443
Rule 2465
Rule 2481
Rule 2495
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x) (i+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(h i-g j) (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(h i-g j) (i+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {h \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {j \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{i+j x} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{e+f x} \, dx}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}-\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {f i-e j}{f}+\frac {j x}{f}\right )}{f i-e j}\right )}{x} \, dx,x,e+f x\right )}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}-\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{f i-e j}\right )}{x} \, dx,x,e+f x\right )}{h i-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}-\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac {2 b^2 p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(652\) vs. \(2(288)=576\).
Time = 0.18 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\frac {a^2 \log (g+h x)-2 a b p q \log (e+f x) \log (g+h x)+b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)+2 a b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+2 a b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-a^2 \log (i+j x)+2 a b p q \log (e+f x) \log (i+j x)-b^2 p^2 q^2 \log ^2(e+f x) \log (i+j x)-2 a b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)+2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)-b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (i+j x)-2 a b p q \log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )-2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )-2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )+2 b^2 p^2 q^2 \operatorname {PolyLog}\left (3,\frac {j (e+f x)}{-f i+e j}\right )}{h i-g j} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right ) \left (j x +i \right )}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right ) \left (i + j x\right )}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (i+j x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{\left (g+h\,x\right )\,\left (i+j\,x\right )} \,d x \]
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