Integrand size = 40, antiderivative size = 141 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {B^2 g (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}-\frac {B g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) i^3 (c+d x)^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d) i^3 (c+d x)^2} \]
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Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2562, 2342, 2341} \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac {B^2 g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
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Rule 2341
Rule 2342
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {g \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) i^3} \\ & = \frac {g (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) i^3 (c+d x)^2}-\frac {(B g) \text {Subst}\left (\int x (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) i^3} \\ & = \frac {B^2 g (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}-\frac {B g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) i^3 (c+d x)^2}+\frac {g (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) i^3 (c+d x)^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 767, normalized size of antiderivative = 5.44 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {g \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-4 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 b B (c+d x) \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b B (c+d x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )-B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 b B (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B \left ((b c-a d)^2+2 b (b c-a d) (c+d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 b^2 B (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 d^2 (b c-a d) i^3 (c+d x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(135)=270\).
Time = 0.90 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.44
method | result | size |
norman | \(\frac {-\frac {2 A^{2} a d g +2 A^{2} b c g -2 B a d g A -2 B b c g A +B^{2} a d g +B^{2} b c g}{4 i \,d^{2}}-\frac {\left (2 A^{2} b g -2 B b g A +B^{2} b g \right ) x}{2 i d}-\frac {B^{2} a^{2} g \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i \left (a d -c b \right )}-\frac {B^{2} b^{2} g \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (a d -c b \right ) i}-\frac {\left (2 A -B \right ) g \,a^{2} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a d -c b \right )}-\frac {B^{2} a b g x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i \left (a d -c b \right )}-\frac {b^{2} g B \left (2 A -B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a d -c b \right )}-\frac {g a B b \left (2 A -B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a d -c b \right )}}{i^{2} \left (d x +c \right )^{2}}\) | \(344\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {2 g \,d^{2} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\) | \(361\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {2 g \,d^{2} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\) | \(361\) |
parallelrisch | \(-\frac {8 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4} g +2 A B \,b^{3} c^{2} d^{2} g +2 A^{2} a^{2} b \,d^{4} g -2 A^{2} b^{3} c^{2} d^{2} g +B^{2} a^{2} b \,d^{4} g -B^{2} b^{3} c^{2} d^{2} g -2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4} g +2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b \,d^{4} g +4 A^{2} x a \,b^{2} d^{4} g -2 A B \,a^{2} b \,d^{4} g +4 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4} g +4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{2} d^{4} g -4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4} g -4 A B x a \,b^{2} d^{4} g +4 A B x \,b^{3} c \,d^{3} g +4 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{4} g -4 A^{2} x \,b^{3} c \,d^{3} g +2 B^{2} x a \,b^{2} d^{4} g -2 B^{2} x \,b^{3} c \,d^{3} g -2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{4} g +2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4} g}{4 i^{3} \left (d x +c \right )^{2} b \,d^{4} \left (a d -c b \right )}\) | \(457\) |
parts | \(\frac {g \,A^{2} \left (-\frac {b}{d^{2} \left (d x +c \right )}-\frac {a d -c b}{2 d^{2} \left (d x +c \right )^{2}}\right )}{i^{3}}-\frac {g \,B^{2} d \left (a \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 )-\frac {c 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}}{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 )}{d}\right )}{i^{3} \left (a d -c b \right )^{2} e^{2}}-\frac {2 g B A d \left (a \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 )-\frac {c 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 )}{d}\right )}{i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(595\) |
risch | \(\text {Expression too large to display}\) | \(2227\) |
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Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (135) = 270\).
Time = 0.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.09 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {2 \, {\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a b d^{2}\right )} g x - 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x + B^{2} a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c^{2} - {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} g - 2 \, {\left ({\left (2 \, A B - B^{2}\right )} b^{2} d^{2} g x^{2} + 2 \, {\left (2 \, A B - B^{2}\right )} a b d^{2} g x + {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (121) = 242\).
Time = 5.09 (sec) , antiderivative size = 712, normalized size of antiderivative = 5.05 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g - \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} + \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} - \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g + \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} - \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} + \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac {\left (- B^{2} a^{2} g - 2 B^{2} a b g x - B^{2} b^{2} g x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a c^{2} d i^{3} + 4 a c d^{2} i^{3} x + 2 a d^{3} i^{3} x^{2} - 2 b c^{3} i^{3} - 4 b c^{2} d i^{3} x - 2 b c d^{2} i^{3} x^{2}} + \frac {- 2 A^{2} a d g - 2 A^{2} b c g + 2 A B a d g + 2 A B b c g - B^{2} a d g - B^{2} b c g + x \left (- 4 A^{2} b d g + 4 A B b d g - 2 B^{2} b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac {\left (- 2 A B a d g - 2 A B b c g - 4 A B b d g x + B^{2} a d g + B^{2} b c g + 2 B^{2} b d g x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1966 vs. \(2 (135) = 270\).
Time = 0.29 (sec) , antiderivative size = 1966, normalized size of antiderivative = 13.94 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\text {Too large to display} \]
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Time = 0.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.40 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (\frac {2 \, {\left (b e x + a e\right )}^{2} B^{2} g \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2} e i^{3}} + \frac {2 \, {\left (2 \, A B g - B^{2} g\right )} {\left (b e x + a e\right )}^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2} e i^{3}} + \frac {{\left (2 \, A^{2} g - 2 \, A B g + B^{2} g\right )} {\left (b e x + a e\right )}^{2}}{{\left (d x + c\right )}^{2} e i^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
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Time = 2.86 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.36 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {x\,\left (2\,b\,d\,g\,A^2-2\,b\,d\,g\,A\,B+b\,d\,g\,B^2\right )+A^2\,a\,d\,g+A^2\,b\,c\,g+\frac {B^2\,a\,d\,g}{2}+\frac {B^2\,b\,c\,g}{2}-A\,B\,a\,d\,g-A\,B\,b\,c\,g}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,a\,g}{2\,d^2\,i^3}+\frac {B^2\,b\,c\,g}{2\,d^3\,i^3}+\frac {B^2\,b\,g\,x}{d^2\,i^3}}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {B^2\,b^2\,g}{2\,d^2\,i^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B\,c\,g}{d^3\,i^3}-x\,\left (\frac {B^2\,g}{d^2\,i^3}-\frac {2\,A\,B\,g}{d^2\,i^3}\right )+\frac {B\,g\,\left (A\,a\,d-B\,a\,d+B\,b\,c\right )}{b\,d^3\,i^3}+\frac {B^2\,b^2\,g\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{d^2\,i^3\,\left (a\,d-b\,c\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,g\,\mathrm {atan}\left (\frac {\left (\frac {2\,a\,d^3\,i^3+2\,b\,c\,d^2\,i^3}{2\,d^2\,i^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A-B\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \]
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