Integrand size = 42, antiderivative size = 375 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {B^2 d^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g i^3} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2562, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i^3 (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {B d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g i^3 (c+d x) (b c-a d)^3}+\frac {4 A b B d (a+b x)}{g i^3 (c+d x) (b c-a d)^3}+\frac {B^2 d^2 (a+b x)^2}{4 g i^3 (c+d x)^2 (b c-a d)^3}+\frac {4 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {4 b B^2 d (a+b x)}{g i^3 (c+d x) (b c-a d)^3} \]
[In]
[Out]
Rule 30
Rule 2332
Rule 2333
Rule 2339
Rule 2341
Rule 2342
Rule 2367
Rule 2388
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))^2}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = \frac {b \text {Subst}\left (\int \frac {(b-d x) (A+B \log (e x))^2}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}-\frac {d \text {Subst}\left (\int (b-d x) (A+B \log (e x))^2 \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = \frac {b^2 \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}-\frac {d \text {Subst}\left (\int \left (b (A+B \log (e x))^2-d x (A+B \log (e x))^2\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}-\frac {(b d) \text {Subst}\left (\int (A+B \log (e x))^2 \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = -\frac {b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \text {Subst}\left (\int x^2 \, dx,x,A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B (b c-a d)^3 g i^3}-\frac {(b d) \text {Subst}\left (\int (A+B \log (e x))^2 \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}+\frac {(2 b B d) \text {Subst}\left (\int (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}+\frac {d^2 \text {Subst}\left (\int x (A+B \log (e x))^2 \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = \frac {2 A b B d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g i^3}+\frac {(2 b B d) \text {Subst}\left (\int (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}+\frac {\left (2 b B^2 d\right ) \text {Subst}\left (\int \log (e x) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3}-\frac {\left (B d^2\right ) \text {Subst}\left (\int x (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = \frac {B^2 d^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {2 b B^2 d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {2 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g i^3}+\frac {\left (2 b B^2 d\right ) \text {Subst}\left (\int \log (e x) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g i^3} \\ & = \frac {B^2 d^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g i^3} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.77 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {\frac {3 \left (2 A^2-2 A B+B^2\right ) (b c-a d)^2}{(c+d x)^2}+\frac {6 b \left (2 A^2-6 A B+7 B^2\right ) (b c-a d)}{c+d x}+6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (a+b x)+\frac {6 B (b c-a d) (B (-7 b c+a d-6 b d x)+A (6 b c-2 a d+4 b d x)) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+\frac {6 B \left (2 A b^2 (c+d x)^2+B d (a+b x) (-4 b c+a d-3 b d x)\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+4 b^2 B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (c+d x)}{12 (b c-a d)^3 g i^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(768\) vs. \(2(367)=734\).
Time = 1.37 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.05
method | result | size |
parts | \(\frac {A^{2} \left (-\frac {1}{2 \left (a d -c b \right ) \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{3}}+\frac {b}{\left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{3}}\right )}{g \,i^{3}}-\frac {B^{2} d \left (\frac {d \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{a d -c b}-\frac {2 b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{a d -c b}+\frac {e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 d \left (a d -c b \right )}\right )}{g \,i^{3} \left (a d -c b \right )^{2} e^{2}}-\frac {2 B A d \left (\frac {d \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{a d -c b}-\frac {2 b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{a d -c b}+\frac {e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 d \left (a d -c b \right )}\right )}{g \,i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(769\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} A^{2} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} A B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {4 d^{3} A B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {2 d^{4} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} B^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} B^{2} b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}\right )}{d^{2}}\) | \(881\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} A^{2} b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} A B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {4 d^{3} A B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {2 d^{4} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} B^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} B^{2} b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}\right )}{d^{2}}\) | \(881\) |
norman | \(\frac {-\frac {2 A^{2} a \,d^{3}-6 A^{2} b c \,d^{2}-2 A B a \,d^{3}+14 A B b c \,d^{2}+B^{2} a \,d^{3}-15 B^{2} b c \,d^{2}}{4 i g \,d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 A^{2} b^{2} c^{2}+2 A B \,a^{2} d^{2}-8 A B a b c d -B^{2} a^{2} d^{2}+8 B^{2} a b c d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {B \left (2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-4 B a b c d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A^{2} b \,d^{2}-6 A B b \,d^{2}+7 B^{2} b \,d^{2}\right ) x}{2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g i}-\frac {B^{2} b^{2} c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (2 A^{2} b^{2} c -2 A B a b d -4 A B \,b^{2} c +3 B^{2} a b d +4 B^{2} b^{2} c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B^{2} d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (2 A^{2} b^{2}-6 A B \,b^{2}+7 B^{2} b^{2}\right ) d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (a d -c b \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b d B \left (2 A b c -B a d -2 B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 d \,B^{2} b^{2} c x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{2} B \,b^{2} \left (2 A -3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{i^{2} \left (d x +c \right )^{2}}\) | \(937\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1122\) |
risch | \(\text {Expression too large to display}\) | \(1661\) |
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Time = 0.30 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.45 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {3 \, {\left (6 \, A^{2} - 14 \, A B + 15 \, B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2} + 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + B^{2} b^{2} c^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 6 \, {\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A B b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2} - 2 \, {\left (B^{2} a b d^{2} - 2 \, {\left (A B - B^{2}\right )} b^{2} c d\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} b^{2} c^{2} - 8 \, {\left (A B - B^{2}\right )} a b c d + {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} + 2 \, {\left (2 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B - 3 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (333) = 666\).
Time = 4.20 (sec) , antiderivative size = 1488, normalized size of antiderivative = 3.97 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (367) = 734\).
Time = 0.34 (sec) , antiderivative size = 2116, normalized size of antiderivative = 5.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \]
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Time = 0.47 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.79 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {1}{12} \, {\left (\frac {4 \, B^{2} b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )^{3}}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + \frac {12 \, A^{2} b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + 6 \, {\left (\frac {2 \, A B b^{2} e}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} - \frac {4 \, {\left (b e x + a e\right )} B^{2} b d}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}} + \frac {{\left (b e x + a e\right )}^{2} B^{2} d^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left (\frac {{\left (2 \, A B d^{2} - B^{2} d^{2}\right )} {\left (b e x + a e\right )}^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (A B b d - B^{2} b d\right )} {\left (b e x + a e\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) + \frac {3 \, {\left (2 \, A^{2} d^{2} - 2 \, A B d^{2} + B^{2} d^{2}\right )} {\left (b e x + a e\right )}^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {24 \, {\left (A^{2} b d - 2 \, A B b d + 2 \, B^{2} b d\right )} {\left (b e x + a e\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
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Time = 4.86 (sec) , antiderivative size = 984, normalized size of antiderivative = 2.62 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,b^2\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B^2\,x\,\left (a\,d-b\,c\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,\left (2\,A-3\,B\right )}{2\,g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\frac {2\,A^2\,a\,d-6\,A^2\,b\,c+B^2\,a\,d-15\,B^2\,b\,c-2\,A\,B\,a\,d+14\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (2\,b\,d\,A^2-6\,b\,d\,A\,B+7\,b\,d\,B^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,a\,d^3\,g\,i^3-2\,b\,c\,d^2\,g\,i^3\right )+x\,\left (4\,a\,c\,d^2\,g\,i^3-4\,b\,c^2\,d\,g\,i^3\right )-2\,b\,c^3\,g\,i^3+2\,a\,c^2\,d\,g\,i^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B^2}{b\,d\,g\,i^3\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )\,\left (2\,A-3\,B\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B\,x\,\left (2\,A-3\,B\right )\,\left (a\,d-b\,c\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}-\frac {B^2\,b^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (A^2-3\,A\,B+\frac {7\,B^2}{2}\right )\,\left (2\,g\,a^3\,d^3\,i^3-2\,g\,a^2\,b\,c\,d^2\,i^3-2\,g\,a\,b^2\,c^2\,d\,i^3+2\,g\,b^3\,c^3\,i^3\right )\,1{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2+7\,B^2\,b^2\right )}+\frac {b^3\,d\,x\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,\left (A^2-3\,A\,B+\frac {7\,B^2}{2}\right )\,4{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2+7\,B^2\,b^2\right )}\right )\,\left (A^2-3\,A\,B+\frac {7\,B^2}{2}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]
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