Integrand size = 32, antiderivative size = 503 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (a+b x)}{6 b d^5}-\frac {13 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}-\frac {2 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {2 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]
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Time = 0.46 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2552, 2356, 2389, 2379, 2438, 2351, 31, 46} \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=-\frac {2 B g^4 (b c-a d)^5 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^5}-\frac {2 B g^4 (c+d x) (b c-a d)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 d^5}+\frac {B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^3}-\frac {2 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{15 b d^2}+\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b}+\frac {2 B^2 g^4 (b c-a d)^5 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac {5 B^2 g^4 (b c-a d)^5 \log (a+b x)}{6 b d^5}-\frac {13 B^2 g^4 (b c-a d)^5 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {13 B^2 g^4 x (b c-a d)^4}{30 d^4}-\frac {7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{60 b d^3}+\frac {B^2 g^4 (a+b x)^3 (b c-a d)^2}{30 b d^2} \]
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2552
Rubi steps \begin{align*} \text {integral}& = -\left (\left ((b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{(d-b x)^6} \, dx,x,\frac {c+d x}{a+b x}\right )\right ) \\ & = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (d-b x)^5} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b} \\ & = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(d-b x)^5} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d} \\ & = \frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^2}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^2}-\frac {\left (B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{10 b d} \\ & = -\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^3}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{15 b d^2}-\frac {\left (B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^4 x}+\frac {b}{d (d-b x)^4}+\frac {b}{d^2 (d-b x)^3}+\frac {b}{d^3 (d-b x)^2}+\frac {b}{d^4 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{10 b d} \\ & = \frac {B^2 (b c-a d)^4 g^4 x}{10 d^4}-\frac {B^2 (b c-a d)^3 g^4 (a+b x)^2}{20 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g^4 \log (a+b x)}{10 b d^5}-\frac {B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{10 b d^5}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^4}+\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (d-b x)} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^4}-\frac {\left (B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^3 x}+\frac {b}{d (d-b x)^3}+\frac {b}{d^2 (d-b x)^2}+\frac {b}{d^3 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{15 b d^2} \\ & = \frac {7 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {7 B^2 (b c-a d)^5 g^4 \log (a+b x)}{30 b d^5}-\frac {7 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}-\frac {2 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {2 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{d-b x} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^5}+\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{b x}\right )}{x} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^5}-\frac {\left (B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^2 x}+\frac {b}{d (d-b x)^2}+\frac {b}{d^2 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3} \\ & = \frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (a+b x)}{6 b d^5}-\frac {13 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}-\frac {2 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {2 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.02 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2-\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x+24 B (b c-a d)^4 \log (a+b x)-4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )-12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+24 b B (b c-a d)^3 (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-24 (b c-a d)^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{12 d^5}\right )}{5 b} \]
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\[\int \left (b g x +a g \right )^{4} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )^{2}d x\]
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\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2395 vs. \(2 (478) = 956\).
Time = 0.31 (sec) , antiderivative size = 2395, normalized size of antiderivative = 4.76 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2 \,d x \]
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