Integrand size = 31, antiderivative size = 285 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {8 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {8 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \]
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Time = 0.33 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2554, 2404, 2354, 2421, 6724} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\frac {4 B \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g}-\frac {4 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g}-\frac {8 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {8 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g} \]
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Rule 2354
Rule 2404
Rule 2421
Rule 2554
Rule 6724
Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right )^2}{(b-d x) (b f-a g-(d f-c g) x)} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = (b c-a d) \text {Subst}\left (\int \left (\frac {d \left (A+B \log \left (e x^2\right )\right )^2}{(b c-a d) g (b-d x)}+\frac {(-d f+c g) \left (A+B \log \left (e x^2\right )\right )^2}{(b c-a d) g (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {d \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right )^2}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}+\frac {((-b c+a d) (d f-c g)) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right )^2}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {(4 B) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right ) \log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}-\frac {(4 B) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right ) \log \left (1+\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {8 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {8 B^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1370\) vs. \(2(285)=570\).
Time = 0.42 (sec) , antiderivative size = 1370, normalized size of antiderivative = 4.81 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\frac {-4 B^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+A^2 \log (f+g x)-4 A B \log \left (\frac {a}{b}+x\right ) \log (f+g x)+4 B^2 \log ^2\left (\frac {a}{b}+x\right ) \log (f+g x)+4 A B \log \left (\frac {c}{d}+x\right ) \log (f+g x)-8 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (f+g x)+4 B^2 \log ^2\left (\frac {c}{d}+x\right ) \log (f+g x)+2 A B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (f+g x)-4 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (f+g x)+4 B^2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (f+g x)+B^2 \log ^2\left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (f+g x)+4 A B \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-4 B^2 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+4 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+8 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-4 B^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+8 B^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-4 B^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-4 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+8 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-4 B^2 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-4 B^2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-8 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+4 B^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-8 B^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+4 B^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {(-b c+a d) (f+g x)}{(d f-c g) (a+b x)}\right )+4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 B \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {g (a+b x)}{-b f+a g}\right )-4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 B \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {g (c+d x)}{-d f+c g}\right )-8 B^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )+8 B^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+8 B^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )-8 B^2 \operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{g} \]
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\[\int \frac {{\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2}}{g x +f}d x\]
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\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
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\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{f+g x} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2}{f+g\,x} \,d x \]
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