Integrand size = 29, antiderivative size = 175 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=-\frac {B (b c-a d)}{(b f-a g) (d f-c g) (f+g x)}+\frac {b^2 B \log (a+b x)}{g (b f-a g)^2}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 g (f+g x)^2}-\frac {B d^2 \log (c+d x)}{g (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2} \]
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Time = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2548, 84} \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{2 g (f+g x)^2}+\frac {b^2 B \log (a+b x)}{g (b f-a g)^2}-\frac {B (b c-a d)}{(f+g x) (b f-a g) (d f-c g)}+\frac {B (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{(b f-a g)^2 (d f-c g)^2}-\frac {B d^2 \log (c+d x)}{g (d f-c g)^2} \]
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Rule 84
Rule 2548
Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 g (f+g x)^2}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (f+g x)^2} \, dx}{g} \\ & = -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 g (f+g x)^2}+\frac {(B (b c-a d)) \int \left (\frac {b^3}{(b c-a d) (b f-a g)^2 (a+b x)}-\frac {d^3}{(b c-a d) (-d f+c g)^2 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^2}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}\right ) \, dx}{g} \\ & = -\frac {B (b c-a d)}{(b f-a g) (d f-c g) (f+g x)}+\frac {b^2 B \log (a+b x)}{g (b f-a g)^2}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 g (f+g x)^2}-\frac {B d^2 \log (c+d x)}{g (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^2}+2 B (b c-a d) \left (\frac {b^2 \log (a+b x)}{(b c-a d) (b f-a g)^2}+\frac {\frac {g (-d f+c g)}{(b f-a g) (f+g x)}+\frac {d^2 \log (c+d x)}{-b c+a d}-\frac {g (-2 b d f+b c g+a d g) \log (f+g x)}{(b f-a g)^2}}{(d f-c g)^2}\right )}{2 g} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(628\) vs. \(2(176)=352\).
Time = 1.22 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.59
method | result | size |
derivativedivides | \(-\frac {-d^{3} A \left (-\frac {1}{\left (c g -d f \right )^{2} \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}-\frac {g}{2 \left (c g -d f \right )^{2} \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )^{2}}\right )+\frac {\frac {B a \,d^{3} g^{2}-B b c \,d^{2} g^{2}}{g^{2} \left (a g -b f \right ) \left (d x +c \right )^{2}}+\frac {b^{2} \left (c g -d f \right ) B d \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{\left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right ) \left (d x +c \right )}-\frac {B a \,d^{3} g^{2}-B b c \,d^{2} g^{2}}{g \left (a c \,g^{2}-a d f g -b c f g +b d \,f^{2}\right ) \left (d x +c \right )}+\frac {B d \left (a^{2} d^{2} g -2 a b \,d^{2} f -b^{2} c^{2} g +2 b^{2} c d f \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 \left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right ) \left (d x +c \right )^{2}}-\frac {b^{2} g B d \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 \left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right )}}{\left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )^{2}}-\frac {B d \left (a^{2} d^{2} g -2 a b \,d^{2} f -b^{2} c^{2} g +2 b^{2} c d f \right ) \ln \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}{a^{2} c^{2} g^{4}-2 a^{2} c d f \,g^{3}+a^{2} d^{2} f^{2} g^{2}-2 a b \,c^{2} f \,g^{3}+4 a b c d \,f^{2} g^{2}-2 a b \,d^{2} f^{3} g +b^{2} c^{2} f^{2} g^{2}-2 b^{2} c d \,f^{3} g +b^{2} d^{2} f^{4}}}{d}\) | \(629\) |
default | \(-\frac {-d^{3} A \left (-\frac {1}{\left (c g -d f \right )^{2} \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}-\frac {g}{2 \left (c g -d f \right )^{2} \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )^{2}}\right )+\frac {\frac {B a \,d^{3} g^{2}-B b c \,d^{2} g^{2}}{g^{2} \left (a g -b f \right ) \left (d x +c \right )^{2}}+\frac {b^{2} \left (c g -d f \right ) B d \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{\left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right ) \left (d x +c \right )}-\frac {B a \,d^{3} g^{2}-B b c \,d^{2} g^{2}}{g \left (a c \,g^{2}-a d f g -b c f g +b d \,f^{2}\right ) \left (d x +c \right )}+\frac {B d \left (a^{2} d^{2} g -2 a b \,d^{2} f -b^{2} c^{2} g +2 b^{2} c d f \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 \left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right ) \left (d x +c \right )^{2}}-\frac {b^{2} g B d \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 \left (g^{2} a^{2}-2 a b f g +f^{2} b^{2}\right )}}{\left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )^{2}}-\frac {B d \left (a^{2} d^{2} g -2 a b \,d^{2} f -b^{2} c^{2} g +2 b^{2} c d f \right ) \ln \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}{a^{2} c^{2} g^{4}-2 a^{2} c d f \,g^{3}+a^{2} d^{2} f^{2} g^{2}-2 a b \,c^{2} f \,g^{3}+4 a b c d \,f^{2} g^{2}-2 a b \,d^{2} f^{3} g +b^{2} c^{2} f^{2} g^{2}-2 b^{2} c d \,f^{3} g +b^{2} d^{2} f^{4}}}{d}\) | \(629\) |
risch | \(\text {Expression too large to display}\) | \(1037\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1586\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (173) = 346\).
Time = 47.45 (sec) , antiderivative size = 1036, normalized size of antiderivative = 5.92 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=-\frac {A b^{2} d^{2} f^{4} + A a^{2} c^{2} g^{4} - 2 \, {\left ({\left (A - B\right )} b^{2} c d + {\left (A + B\right )} a b d^{2}\right )} f^{3} g + {\left ({\left (A - 2 \, B\right )} b^{2} c^{2} + 4 \, A a b c d + {\left (A + 2 \, B\right )} a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left ({\left (A - B\right )} a b c^{2} + {\left (A + B\right )} a^{2} c d\right )} f g^{3} + 2 \, {\left ({\left (B b^{2} c d - B a b d^{2}\right )} f^{2} g^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f g^{3} + {\left (B a b c^{2} - B a^{2} c d\right )} g^{4}\right )} x - 2 \, {\left (B b^{2} d^{2} f^{4} - 2 \, B b^{2} c d f^{3} g + B b^{2} c^{2} f^{2} g^{2} + {\left (B b^{2} d^{2} f^{2} g^{2} - 2 \, B b^{2} c d f g^{3} + B b^{2} c^{2} g^{4}\right )} x^{2} + 2 \, {\left (B b^{2} d^{2} f^{3} g - 2 \, B b^{2} c d f^{2} g^{2} + B b^{2} c^{2} f g^{3}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B b^{2} d^{2} f^{4} - 2 \, B a b d^{2} f^{3} g + B a^{2} d^{2} f^{2} g^{2} + {\left (B b^{2} d^{2} f^{2} g^{2} - 2 \, B a b d^{2} f g^{3} + B a^{2} d^{2} g^{4}\right )} x^{2} + 2 \, {\left (B b^{2} d^{2} f^{3} g - 2 \, B a b d^{2} f^{2} g^{2} + B a^{2} d^{2} f g^{3}\right )} x\right )} \log \left (d x + c\right ) - 2 \, {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f^{3} g - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f^{2} g^{2} + {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f g^{3} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g^{4}\right )} x^{2} + 2 \, {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} f^{2} g^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} f g^{3}\right )} x\right )} \log \left (g x + f\right ) + {\left (B b^{2} d^{2} f^{4} + B a^{2} c^{2} g^{4} - 2 \, {\left (B b^{2} c d + B a b d^{2}\right )} f^{3} g + {\left (B b^{2} c^{2} + 4 \, B a b c d + B a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left (B a b c^{2} + B a^{2} c d\right )} f g^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (b^{2} d^{2} f^{6} g + a^{2} c^{2} f^{2} g^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{5} g^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{4} g^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3} g^{4} + {\left (b^{2} d^{2} f^{4} g^{3} + a^{2} c^{2} g^{7} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g^{4} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{5} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{6}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} f^{5} g^{2} + a^{2} c^{2} f g^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{4} g^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3} g^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} g^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (173) = 346\).
Time = 0.23 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.31 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=\frac {1}{2} \, {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{2} f^{2} g - 2 \, a b f g^{2} + a^{2} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{d^{2} f^{2} g - 2 \, c d f g^{2} + c^{2} g^{3}} + \frac {2 \, {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} + a^{2} c^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{3}} - \frac {2 \, {\left (b c - a d\right )}}{b d f^{3} + a c f g^{2} - {\left (b c + a d\right )} f^{2} g + {\left (b d f^{2} g + a c g^{3} - {\left (b c + a d\right )} f g^{2}\right )} x} - \frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g}\right )} B - \frac {A}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (173) = 346\).
Time = 0.50 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.78 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=\frac {B b^{3} \log \left ({\left | b x + a \right |}\right )}{b^{3} f^{2} g - 2 \, a b^{2} f g^{2} + a^{2} b g^{3}} - \frac {B d^{3} \log \left ({\left | d x + c \right |}\right )}{d^{3} f^{2} g - 2 \, c d^{2} f g^{2} + c^{2} d g^{3}} + \frac {{\left (2 \, B b^{2} c d f - 2 \, B a b d^{2} f - B b^{2} c^{2} g + B a^{2} d^{2} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} - 2 \, b^{2} c d f^{3} g - 2 \, a b d^{2} f^{3} g + b^{2} c^{2} f^{2} g^{2} + 4 \, a b c d f^{2} g^{2} + a^{2} d^{2} f^{2} g^{2} - 2 \, a b c^{2} f g^{3} - 2 \, a^{2} c d f g^{3} + a^{2} c^{2} g^{4}} - \frac {B \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {2 \, B b c g^{2} x - 2 \, B a d g^{2} x + A b d f^{2} - A b c f g + 2 \, B b c f g - A a d f g - 2 \, B a d f g + A a c g^{2}}{2 \, {\left (b d f^{2} g^{3} x^{2} - b c f g^{4} x^{2} - a d f g^{4} x^{2} + a c g^{5} x^{2} + 2 \, b d f^{3} g^{2} x - 2 \, b c f^{2} g^{3} x - 2 \, a d f^{2} g^{3} x + 2 \, a c f g^{4} x + b d f^{4} g - b c f^{3} g^{2} - a d f^{3} g^{2} + a c f^{2} g^{3}\right )}} \]
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Time = 3.97 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.35 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3} \, dx=\frac {\ln \left (f+g\,x\right )\,\left (g\,\left (B\,a^2\,d^2-B\,b^2\,c^2\right )-2\,B\,a\,b\,d^2\,f+2\,B\,b^2\,c\,d\,f\right )}{a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4}-\frac {\frac {A\,a\,c\,g^2+A\,b\,d\,f^2-A\,a\,d\,f\,g-A\,b\,c\,f\,g-2\,B\,a\,d\,f\,g+2\,B\,b\,c\,f\,g}{2\,\left (a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g\right )}-\frac {x\,\left (B\,a\,d\,g^2-B\,b\,c\,g^2\right )}{a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g}}{f^2\,g+2\,f\,g^2\,x+g^3\,x^2}+\frac {B\,b^2\,\ln \left (a+b\,x\right )}{a^2\,g^3-2\,a\,b\,f\,g^2+b^2\,f^2\,g}-\frac {B\,d^2\,\ln \left (c+d\,x\right )}{c^2\,g^3-2\,c\,d\,f\,g^2+d^2\,f^2\,g}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{2\,g\,\left (f^2+2\,f\,g\,x+g^2\,x^2\right )} \]
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