Integrand size = 42, antiderivative size = 183 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {2 b B^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g^2 i} \]
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Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2562, 2395, 2342, 2341, 2339, 30} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^2 i (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B^2 (c+d x)}{g^2 i (a+b x) (b c-a d)^2} \]
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Rule 30
Rule 2339
Rule 2341
Rule 2342
Rule 2395
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x) (A+B \log (e x))^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b (A+B \log (e x))^2}{x^2}-\frac {d (A+B \log (e x))^2}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i} \\ & = \frac {b \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i}-\frac {d \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)^2 g^2 i} \\ & = -\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}+\frac {(2 b B) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^2 i}-\frac {d \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)^2 g^2 i} \\ & = -\frac {2 b B^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g^2 i} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {3 \left (A^2+2 A B+2 B^2\right ) d (a+b x) \log (a+b x)+6 B (A+B) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (a A d+A b d x+b B (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 d (a+b x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )+3 \left (A^2+2 A B+2 B^2\right ) (b c-a d-d (a+b x) \log (c+d x))}{3 (b c-a d)^2 g^2 i (a+b x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(181)=362\).
Time = 0.98 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.19
method | result | size |
norman | \(\frac {-\frac {\left (A^{2} a d +2 A B b c +2 B^{2} b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B \left (A a d +B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} a d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b \left (A^{2} d +2 A B d +2 B^{2} d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (A^{2}+2 B A +2 B^{2}\right ) b x}{g i a \left (a d -c b \right )}-\frac {b \,B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B b d \left (A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g \left (b x +a \right )}\) | \(400\) |
parts | \(\frac {A^{2} \left (\frac {d \ln \left (d x +c \right )}{\left (a d -c b \right )^{2}}+\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}-\frac {d \ln \left (b x +a \right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i}-\frac {B^{2} \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (a d -c b \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i d}-\frac {2 B A \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i d}\) | \(460\) |
parallelrisch | \(-\frac {3 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +6 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{4} c^{2} d +3 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{3}-3 A^{2} x \,a^{2} b^{2} c^{3}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} c^{2} d -6 B^{2} x \,a^{2} b^{2} c^{3}+6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} c^{2} d +3 A^{2} x \,a^{3} b \,c^{2} d -6 A B x \,a^{2} b^{2} c^{3}+6 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+6 B^{2} x \,a^{3} b \,c^{2} d +B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{3} b \,c^{2} d +3 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +3 A^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 A B x \,a^{3} b \,c^{2} d}{3 i \,g^{2} \left (b x +a \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) c^{2} a^{3}}\) | \(473\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\) | \(527\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\) | \(527\) |
risch | \(\frac {A^{2} d \ln \left (d x +c \right )}{g^{2} i \left (a d -c b \right )^{2}}+\frac {A^{2}}{g^{2} i \left (b x +a \right ) \left (a d -c b \right )}-\frac {A^{2} d \ln \left (b x +a \right )}{g^{2} i \left (a d -c b \right )^{2}}-\frac {B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g^{2} i \left (a d -c b \right )^{2}}-\frac {B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B A d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (a d -c b \right )^{2}}-\frac {2 B A b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B A b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\) | \(556\) |
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Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.26 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {{\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (B^{2} b c + A B a d + {\left (A B + B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} a d + {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b d x + 2 \, {\left (A B + B^{2}\right )} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (158) = 316\).
Time = 0.74 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.96 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=- \frac {B^{2} d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g^{2} i - 6 a b c d g^{2} i + 3 b^{2} c^{2} g^{2} i} + \frac {\left (2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A^{2} + 2 A B + 2 B^{2}\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) + \frac {\left (- A B a d - A B b d x - B^{2} b c - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{2} g^{2} i - 2 a^{2} b c d g^{2} i + a^{2} b d^{2} g^{2} i x + a b^{2} c^{2} g^{2} i - 2 a b^{2} c d g^{2} i x + b^{3} c^{2} g^{2} i x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (181) = 362\).
Time = 0.26 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.51 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-B^{2} {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} - 2 \, A B {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left ({\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x} - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{3} - {\left (b d x + a d\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (b d x + a d\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, b c - 6 \, a d + 6 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d x + {\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x}\right )} - A^{2} {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} + \frac {{\left ({\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x} \]
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\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}} \,d x } \]
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Time = 2.97 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.29 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\frac {A^2+2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,d\,\left (A+B\right )}{g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )\,\left (A+B\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {d\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (d\,A^2+2\,d\,A\,B+2\,d\,B^2\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \]
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