Integrand size = 47, antiderivative size = 137 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B n (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{(b c-a d) i^2 (1+m)^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 (1+m) (c+d x)} \]
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Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2563, 2341} \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1) (c+d x) (b c-a d)}-\frac {B n (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{i^2 (m+1)^2 (c+d x) (b c-a d)} \]
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Rule 2341
Rule 2563
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((g (a+b x))^{-2-m} \left (\frac {a+b x}{c+d x}\right )^{2+m} (i (c+d x))^{2+m}\right ) \text {Subst}\left (\int x^{-2-m} \left (A+B \log \left (e x^n\right )\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) i^2} \\ & = -\frac {B n (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{(b c-a d) i^2 (1+m)^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 (1+m) (c+d x)} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {(g (a+b x))^{-1-m} (c+d x) (i (c+d x))^m \left (A+A m+B n+B (1+m) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) g (1+m)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(820\) vs. \(2(137)=274\).
Time = 31.01 (sec) , antiderivative size = 821, normalized size of antiderivative = 5.99
method | result | size |
parallelrisch | \(\frac {B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a b \,d^{2} m n +B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{2} c d m n +B \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a b c d m n +B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{2} c d n +A \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b c d m n +B \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a b c d n +A \,x^{2} \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} d^{2} n +B \,x^{2} \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{2} d^{2} m n +A x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b \,d^{2} m n +A x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} c d m n +B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a b \,d^{2} n +B \,x^{2} \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} d^{2} n^{2}+A \,x^{2} \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} d^{2} m n +B \,x^{2} \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{2} d^{2} n +B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b \,d^{2} n^{2}+B x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} c d \,n^{2}+A x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b \,d^{2} n +A x \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} b^{2} c d n +B \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b c d \,n^{2}+A \left (i \left (d x +c \right )\right )^{m} \left (g \left (b x +a \right )\right )^{-2-m} a b c d n}{d b n \left (a d m -b c m +a d -c b \right ) \left (1+m \right )}\) | \(821\) |
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Time = 0.37 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.96 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {{\left (A a c m + B a c n + A a c + {\left (A b d m + B b d n + A b d\right )} x^{2} + {\left (A b c + A a d + {\left (A b c + A a d\right )} m + {\left (B b c + B a d\right )} n\right )} x + {\left (B a c m + B a c + {\left (B b d m + B b d\right )} x^{2} + {\left (B b c + B a d + {\left (B b c + B a d\right )} m\right )} x\right )} \log \left (e\right ) + {\left ({\left (B b d m + B b d\right )} n x^{2} + {\left (B b c + B a d + {\left (B b c + B a d\right )} m\right )} n x + {\left (B a c m + B a c\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac {b x + a}{d x + c}\right ) + m \log \left (\frac {i}{g}\right )\right )}}{{\left (b c - a d\right )} m^{2} + b c - a d + 2 \, {\left (b c - a d\right )} m} \]
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Exception generated. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \]
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\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \]
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Timed out. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^m\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \]
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