Optimal. Leaf size=132 \[ -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]
[Out]
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Rubi [A]
time = 0.11, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2550, 2379,
2421, 6724} \begin {gather*} \frac {4 B \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g}+\frac {8 B^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b g}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2379
Rule 2421
Rule 2550
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{a g+b g x} \, dx &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{e (a+b x)^2} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{(a+b x)^2} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {2 (b c-a d) e \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B (b c-a d)) \int \left (\frac {d \left (-A-B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B) \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(4 B d) \int \frac {\left (-A-B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B) \int \left (\frac {A \log (a g+b g x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{g}-\frac {(4 B d) \int \left (\frac {A \log (a g+b g x)}{-c-d x}+\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 A B) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (4 B^2\right ) \int \frac {\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(4 A B d) \int \frac {\log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-(4 A B) \int \frac {\log \left (\frac {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx-\frac {(4 A B) \text {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac {\left (4 B^2\right ) \int \frac {\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left ((a+b x)^2\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left (\frac {1}{(c+d x)^2}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (4 B^2 d \left (-\log \left ((a+b x)^2\right )-\log \left (\frac {1}{(c+d x)^2}\right )+\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )\right ) \int \frac {\log (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\left (8 B^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\frac {(4 A B) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}-\frac {(4 A B) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^2\right ) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x^2}\right ) \log \left (\frac {-b c g+a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\left (4 B^2 \left (-\log \left ((a+b x)^2\right )-\log \left (\frac {1}{(c+d x)^2}\right )+\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )\right ) \int \frac {\log \left (\frac {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x^2}\right )}{\frac {-b c g+a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{\frac {-b c g+a d g}{d}-\frac {b g x}{d}} \, dx,x,-c-d x\right )}{d}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b g \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^2\right )}{\frac {-b c+a d}{d}-\frac {b x}{d}} \, dx,x,-c-d x\right )}{d g}-\frac {\left (4 B^2 \left (-\log \left ((a+b x)^2\right )-\log \left (\frac {1}{(c+d x)^2}\right )+\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\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-a d}\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}-\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \log \left (\frac {-b c+a d}{b}-\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x^2}\right ) \log \left (1+\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{b g}-\frac {d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\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-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(c+d x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {\left (4 B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c g+a d g}{b g}-\frac {d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^2\right )}{\frac {-b c+a d}{b}-\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 g}\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\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-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(c+d x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \log \left (1-\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((a+b x)^2\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\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-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(c+d x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {4 B^2 \log (g (a+b x)) \log \left ((a+b x)^2\right ) \log (-c-d x)}{b g}-\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{(c+d x)^2}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2\left ((a+b x)^2\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {2 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((a+b x)^2\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \left (\log \left ((a+b x)^2\right )+\log \left (\frac {1}{(c+d x)^2}\right )-\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-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(c+d x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 257, normalized size = 1.95 \begin {gather*} \frac {2 A B \log ^2\left (\frac {a}{b}+x\right )+A^2 \log (a+b x)-4 A B \log \left (\frac {a}{b}+x\right ) \log (a+b x)+4 A B \log \left (\frac {c}{d}+x\right ) \log (a+b x)-4 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 A B \log (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-B^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-4 A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+4 B^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )+8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )^{2}}{b g x +a g}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A^{2}}{a + b x}\, dx + \int \frac {B^{2} \log {\left (\frac {a^{2} e}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {2 a b e x}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {b^{2} e x^{2}}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B \log {\left (\frac {a^{2} e}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {2 a b e x}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {b^{2} e x^{2}}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}}{a + b x}\, dx}{g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2}{a\,g+b\,g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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