Integrand size = 32, antiderivative size = 120 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {2 B (b c-a d)^2 g^2 x}{3 d^2}-\frac {B (b c-a d) g^2 (a+b x)^2}{3 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b}-\frac {2 B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2548, 21, 45} \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b}-\frac {2 B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {2 B g^2 x (b c-a d)^2}{3 d^2}-\frac {B g^2 (a+b x)^2 (b c-a d)}{3 b d} \]
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Rule 21
Rule 45
Rule 2548
Rubi steps \begin{align*} \text {integral}& = \frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b}-\frac {(2 B (b c-a d)) \int \frac {(a g+b g x)^3}{(a+b x) (c+d x)} \, dx}{3 b g} \\ & = \frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b}-\frac {\left (2 B (b c-a d) g^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b} \\ & = \frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b}-\frac {\left (2 B (b c-a d) g^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b} \\ & = \frac {2 B (b c-a d)^2 g^2 x}{3 d^2}-\frac {B (b c-a d) g^2 (a+b x)^2}{3 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b}-\frac {2 B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+\frac {B (-b c+a d) \left (d \left (a^2 d+4 a b d x+b^2 x (-2 c+d x)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{d^3}\right )}{3 b} \]
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Time = 0.70 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.75
method | result | size |
risch | \(\frac {\left (b x +a \right )^{3} g^{2} B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{3 b}+\frac {g^{2} b^{2} A \,x^{3}}{3}+g^{2} b A a \,x^{2}+\frac {g^{2} b B a \,x^{2}}{3}-\frac {g^{2} b^{2} B c \,x^{2}}{3 d}+g^{2} A \,a^{2} x +\frac {2 g^{2} B \ln \left (d x +c \right ) a^{3}}{3 b}-\frac {2 g^{2} B \ln \left (d x +c \right ) a^{2} c}{d}+\frac {2 g^{2} b B \ln \left (d x +c \right ) a \,c^{2}}{d^{2}}-\frac {2 g^{2} b^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}+\frac {4 g^{2} B \,a^{2} x}{3}-\frac {2 g^{2} b B a c x}{d}+\frac {2 g^{2} b^{2} B \,c^{2} x}{3 d^{2}}\) | \(210\) |
parts | \(\frac {A \,g^{2} \left (b x +a \right )^{3}}{3 b}-\frac {B \,g^{2} \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}+\frac {\ln \left (\frac {1}{d x +c}\right )}{b}\right )\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}-\frac {\left (d x +c \right )^{2}}{2 b}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}\right )\right ) b^{2}+2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}\right )\right ) b \left (a d -c b \right )\right )}{d^{3}}\) | \(447\) |
parallelrisch | \(\frac {-12 B \ln \left (b x +a \right ) a^{2} b c \,d^{2} g^{2}+12 B \ln \left (b x +a \right ) a \,b^{2} c^{2} d \,g^{2}+6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{2} d^{3} g^{2}+6 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b \,d^{3} g^{2}+10 B a \,c^{2} d \,g^{2} b^{2}-6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{2} c^{2} d \,g^{2}+2 A \,x^{3} b^{3} d^{3} g^{2}+2 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c^{3} g^{2}+4 B \ln \left (b x +a \right ) a^{3} d^{3} g^{2}-4 B \ln \left (b x +a \right ) b^{3} c^{3} g^{2}-12 A \,a^{2} b c \,d^{2} g^{2}+2 B \,a^{2} b c \,d^{2} g^{2}-2 B \,x^{2} b^{3} c \,d^{2} g^{2}+6 A x \,a^{2} b \,d^{3} g^{2}+8 B x \,a^{2} b \,d^{3} g^{2}-4 B \,c^{3} g^{2} b^{3}-6 A \,a^{3} d^{3} g^{2}-8 B \,a^{3} d^{3} g^{2}+4 B x \,b^{3} c^{2} d \,g^{2}+2 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{3} g^{2}+6 A \,x^{2} a \,b^{2} d^{3} g^{2}+2 B \,x^{2} a \,b^{2} d^{3} g^{2}-12 B x a \,b^{2} c \,d^{2} g^{2}+6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b c \,d^{2} g^{2}}{6 b \,d^{3}}\) | \(471\) |
derivativedivides | \(-\frac {\frac {A \,g^{2} \left (-\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right )-\frac {b^{2} \left (d x +c \right )^{3}}{3}-b \left (a d -c b \right ) \left (d x +c \right )^{2}\right )}{d^{2}}+\frac {B \,g^{2} \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}+\frac {\ln \left (\frac {1}{d x +c}\right )}{b}\right )\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}-\frac {\left (d x +c \right )^{2}}{2 b}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}\right )\right ) b^{2}+2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}\right )\right ) b \left (a d -c b \right )\right )}{d^{2}}}{d}\) | \(502\) |
default | \(-\frac {\frac {A \,g^{2} \left (-\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right )-\frac {b^{2} \left (d x +c \right )^{3}}{3}-b \left (a d -c b \right ) \left (d x +c \right )^{2}\right )}{d^{2}}+\frac {B \,g^{2} \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}+\frac {\ln \left (\frac {1}{d x +c}\right )}{b}\right )\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}-\frac {\left (d x +c \right )^{2}}{2 b}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}\right )\right ) b^{2}+2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}\right )\right ) b \left (a d -c b \right )\right )}{d^{2}}}{d}\) | \(502\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (112) = 224\).
Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.02 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) - {\left (B b^{3} c d^{2} - {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} + {\left (2 \, B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{2} b d^{3}\right )} g^{2} x - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, b d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (107) = 214\).
Time = 1.39 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.31 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {A b^{2} g^{2} x^{3}}{3} + \frac {2 B a^{3} g^{2} \log {\left (x + \frac {\frac {2 B a^{4} d^{3} g^{2}}{b} + 6 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 b} - \frac {2 B c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {8 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2} - 2 B a c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {2 B b c^{2} g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} + \frac {B a b g^{2}}{3} - \frac {B b^{2} c g^{2}}{3 d}\right ) + x \left (A a^{2} g^{2} + \frac {4 B a^{2} g^{2}}{3} - \frac {2 B a b c g^{2}}{d} + \frac {2 B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (112) = 224\).
Time = 0.36 (sec) , antiderivative size = 437, normalized size of antiderivative = 3.64 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{3} \, {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (112) = 224\).
Time = 1.81 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.05 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + \frac {2 \, B a^{3} g^{2} \log \left (b x + a\right )}{3 \, b} - \frac {{\left (B b^{2} c g^{2} - 3 \, A a b d g^{2} - B a b d g^{2}\right )} x^{2}}{3 \, d} + \frac {1}{3} \, {\left (B b^{2} g^{2} x^{3} + 3 \, B a b g^{2} x^{2} + 3 \, B a^{2} g^{2} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {{\left (2 \, B b^{2} c^{2} g^{2} - 6 \, B a b c d g^{2} + 3 \, A a^{2} d^{2} g^{2} + 4 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} - \frac {2 \, {\left (B b^{2} c^{3} g^{2} - 3 \, B a b c^{2} d g^{2} + 3 \, B a^{2} c d^{2} g^{2}\right )} \log \left (-d x - c\right )}{3 \, d^{3}} \]
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Time = 1.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.47 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,a^2\,c\,d^2\,g^2-6\,B\,a\,b\,c^2\,d\,g^2+2\,B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {2\,B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \]
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