Integrand size = 34, antiderivative size = 299 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {4 A B d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}+\frac {8 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{(b c-a d)^2 g^3 (a+b x)^2}-\frac {4 B^2 d (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(b c-a d)^2 g^3 (a+b x)}+\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2552, 2367, 2333, 2332, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {b B (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{g^3 (a+b x)^2 (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac {4 A B d (c+d x)}{g^3 (a+b x) (b c-a d)^2}-\frac {4 B^2 d (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{g^3 (a+b x) (b c-a d)^2}-\frac {b B^2 (c+d x)^2}{g^3 (a+b x)^2 (b c-a d)^2}+\frac {8 B^2 d (c+d x)}{g^3 (a+b x) (b c-a d)^2} \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2367
Rule 2552
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (d-b x) \left (A+B \log \left (e x^2\right )\right )^2 \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {\text {Subst}\left (\int \left (d \left (A+B \log \left (e x^2\right )\right )^2-b x \left (A+B \log \left (e x^2\right )\right )^2\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3} \\ & = -\frac {b \text {Subst}\left (\int x \left (A+B \log \left (e x^2\right )\right )^2 \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3}+\frac {d \text {Subst}\left (\int \left (A+B \log \left (e x^2\right )\right )^2 \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {d (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2}+\frac {(2 b B) \text {Subst}\left (\int x \left (A+B \log \left (e x^2\right )\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3}-\frac {(4 B d) \text {Subst}\left (\int \left (A+B \log \left (e x^2\right )\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3} \\ & = -\frac {4 A B d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2}-\frac {\left (4 B^2 d\right ) \text {Subst}\left (\int \log \left (e x^2\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d)^2 g^3} \\ & = -\frac {4 A B d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}+\frac {8 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{(b c-a d)^2 g^3 (a+b x)^2}-\frac {4 B^2 d (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(b c-a d)^2 g^3 (a+b x)}+\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.28 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.51 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {2 B \left (4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+(b c-a d)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+2 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+2 d^2 (a+b x)^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^2}}{2 b g^3 (a+b x)^2} \]
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Time = 1.42 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.64
method | result | size |
norman | \(\frac {\frac {\left (A^{2} a d -A^{2} b c -4 A B a d +2 A B b c +8 B^{2} a d -2 B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B c \left (2 A a d -A b c -4 B a d +B b c \right ) \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b B \,d^{2} \left (A -3 B \right ) x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (A^{2} a d -A^{2} b c -6 A B a d +2 A B b c +14 B^{2} a d -2 B^{2} b c \right ) b \,x^{2}}{2 a^{2} g \left (a d -c b \right )}+\frac {2 B d \left (A a d -2 B a d -B b c \right ) x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,B^{2} d^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}}{g^{2} \left (b x +a \right )^{2}}\) | \(490\) |
derivativedivides | \(-\frac {\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2}}+\frac {B^{2}}{g^{3} \left (b x +a \right )^{2}}-\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (b x +a \right )^{2}}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{2 g^{3} \left (b x +a \right )^{2}}+\frac {6 B^{2} d}{g^{3} \left (a d -c b \right ) \left (b x +a \right )}+\frac {3 B^{2} d^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} d^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{2 g^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (a d -c b \right ) \left (b x +a \right )}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2 \left (b x +a \right )^{2}}-\left (a d -c b \right ) \left (\frac {\frac {a d}{2 \left (b x +a \right )^{2}}-\frac {b c}{2 \left (b x +a \right )^{2}}+\frac {d}{b x +a}}{\left (a d -c b \right )^{2}}+\frac {d^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{3}}\right )\right )}{g^{3}}}{b}\) | \(499\) |
default | \(-\frac {\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2}}+\frac {B^{2}}{g^{3} \left (b x +a \right )^{2}}-\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (b x +a \right )^{2}}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{2 g^{3} \left (b x +a \right )^{2}}+\frac {6 B^{2} d}{g^{3} \left (a d -c b \right ) \left (b x +a \right )}+\frac {3 B^{2} d^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} d^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{2 g^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} \left (a d -c b \right ) \left (b x +a \right )}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2 \left (b x +a \right )^{2}}-\left (a d -c b \right ) \left (\frac {\frac {a d}{2 \left (b x +a \right )^{2}}-\frac {b c}{2 \left (b x +a \right )^{2}}+\frac {d}{b x +a}}{\left (a d -c b \right )^{2}}+\frac {d^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{3}}\right )\right )}{g^{3}}}{b}\) | \(499\) |
parts | \(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {b \left (7 B^{2} a d -B^{2} b c \right ) x^{2}}{a^{2} g \left (a d -c b \right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (4 B^{2} a d -B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (4 a d -c b \right ) B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )^{2}}-\frac {3 b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,B^{2} d^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}-\frac {2 \left (2 a d +c b \right ) B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}-\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2 \left (b x +a \right )^{2}}-\left (a d -c b \right ) \left (\frac {\frac {a d}{2 \left (b x +a \right )^{2}}-\frac {b c}{2 \left (b x +a \right )^{2}}+\frac {d}{b x +a}}{\left (a d -c b \right )^{2}}+\frac {d^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{3}}\right )\right )}{g^{3} b}\) | \(578\) |
parallelrisch | \(-\frac {-2 A^{2} a \,b^{4} c \,d^{2}-6 A B \,a^{2} b^{3} d^{3}-2 A B \,b^{5} c^{2} d -16 B^{2} a \,b^{4} c \,d^{2}-2 B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{5} c^{2} d -4 A B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{4} d^{3}-4 A B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{4} c \,d^{2}+8 A B a \,b^{4} c \,d^{2}-4 A B x a \,b^{4} d^{3}+4 A B x \,b^{5} c \,d^{2}+12 B^{2} x a \,b^{4} d^{3}-12 B^{2} x \,b^{5} c \,d^{2}+2 A B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{5} c^{2} d +8 B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{4} c \,d^{2}-2 A B \,x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{5} d^{3}-2 B^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} a \,b^{4} d^{3}+8 B^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{4} d^{3}+4 B^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{5} c \,d^{2}-2 B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} a \,b^{4} c \,d^{2}-B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} b^{5} d^{3}+6 B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{5} d^{3}+B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} b^{5} c^{2} d +A^{2} a^{2} b^{3} d^{3}+A^{2} b^{5} c^{2} d +14 B^{2} a^{2} b^{3} d^{3}+2 B^{2} b^{5} c^{2} d}{2 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} d}\) | \(598\) |
risch | \(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {b \left (7 B^{2} a d -B^{2} b c \right ) x^{2}}{a^{2} g \left (a d -c b \right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (4 B^{2} a d -B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (4 a d -c b \right ) B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )^{2}}-\frac {3 b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,B^{2} d^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}-\frac {2 \left (2 a d +c b \right ) B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}-\frac {A B \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{3} b \left (b x +a \right )^{2}}+\frac {A B \,a^{2} d^{2}}{g^{3} b \left (a d -c b \right )^{2} \left (b x +a \right )^{2}}-\frac {2 A B a d c}{g^{3} \left (a d -c b \right )^{2} \left (b x +a \right )^{2}}+\frac {A B b \,c^{2}}{g^{3} \left (a d -c b \right )^{2} \left (b x +a \right )^{2}}+\frac {2 A B \,d^{2} a}{g^{3} b \left (a d -c b \right )^{2} \left (b x +a \right )}-\frac {2 A B d c}{g^{3} \left (a d -c b \right )^{2} \left (b x +a \right )}+\frac {2 A B \,d^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right ) a}{g^{3} b \left (a d -c b \right )^{3}}-\frac {2 A B \,d^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right ) c}{g^{3} \left (a d -c b \right )^{3}}\) | \(721\) |
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Time = 0.28 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.38 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} - 2 \, {\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a b c d + {\left (A^{2} - 6 \, A B + 14 \, B^{2}\right )} a^{2} d^{2} - {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 4 \, {\left ({\left (A B - 3 \, B^{2}\right )} b^{2} c d - {\left (A B - 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (A B - B^{2}\right )} b^{2} c^{2} + 2 \, {\left (A B - 2 \, B^{2}\right )} a b c d - 2 \, {\left (B^{2} b^{2} c d - {\left (A B - 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{2 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (279) = 558\).
Time = 2.19 (sec) , antiderivative size = 877, normalized size of antiderivative = 2.93 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {2 B d^{2} \left (A - 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} - 6 B^{2} a d^{3} - 6 B^{2} b c d^{2} - \frac {2 B a^{3} d^{5} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a^{2} b c d^{4} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a b^{2} c^{2} d^{3} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {2 B b^{3} c^{3} d^{2} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} - 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} - \frac {2 B d^{2} \left (A - 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} - 6 B^{2} a d^{3} - 6 B^{2} b c d^{2} + \frac {2 B a^{3} d^{5} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a^{2} b c d^{4} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a b^{2} c^{2} d^{3} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {2 B b^{3} c^{3} d^{2} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} - 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {\left (2 B^{2} a c d + 2 B^{2} a d^{2} x - B^{2} b c^{2} + B^{2} b d^{2} x^{2}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}^{2}}{2 a^{4} d^{2} g^{3} - 4 a^{3} b c d g^{3} + 4 a^{3} b d^{2} g^{3} x + 2 a^{2} b^{2} c^{2} g^{3} - 8 a^{2} b^{2} c d g^{3} x + 2 a^{2} b^{2} d^{2} g^{3} x^{2} + 4 a b^{3} c^{2} g^{3} x - 4 a b^{3} c d g^{3} x^{2} + 2 b^{4} c^{2} g^{3} x^{2}} + \frac {\left (- A B a d + A B b c + 3 B^{2} a d - B^{2} b c + 2 B^{2} b d x\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}}{a^{3} b d g^{3} - a^{2} b^{2} c g^{3} + 2 a^{2} b^{2} d g^{3} x - 2 a b^{3} c g^{3} x + a b^{3} d g^{3} x^{2} - b^{4} c g^{3} x^{2}} + \frac {- A^{2} a d + A^{2} b c + 6 A B a d - 2 A B b c - 14 B^{2} a d + 2 B^{2} b c + x \left (4 A B b d - 12 B^{2} b d\right )}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (297) = 594\).
Time = 0.28 (sec) , antiderivative size = 1001, normalized size of antiderivative = 3.35 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-{\left ({\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} - A B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {\log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {B^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
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\[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3}} \,d x } \]
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Time = 3.33 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.69 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (\frac {2\,B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {A\,B}{b^2\,d\,g^3}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-{\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {A^2\,a\,d-A^2\,b\,c+14\,B^2\,a\,d-2\,B^2\,b\,c-6\,A\,B\,a\,d+2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {2\,x\,\left (3\,B^2\,b\,d-A\,B\,b\,d\right )}{a\,d-b\,c}}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (A-3\,B\right )\,2{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (6\,B^2\,d^2-2\,A\,B\,d^2\right )}\right )\,\left (A-3\,B\right )\,4{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]
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