Integrand size = 77, antiderivative size = 101 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {97} \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{m+1} (c+d x)^{-\frac {d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac {f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]
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Rule 97
Rubi steps \begin{align*} \text {integral}& = \frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \]
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Time = 39.86 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.60
method | result | size |
gosper | \(\frac {b \left (b x +a \right )^{1+m} \left (d x +c \right )^{1-\frac {a d f m -b d e m +a d f +b c f -2 b d e}{b \left (c f -d e \right )}} \left (f x +e \right )^{1+\frac {a d f m -b c f m +a d f -2 b c f +b d e}{b \left (c f -d e \right )}}}{a^{2} d f m -a b c f m -a b d e m +b^{2} c e m +a^{2} d f -a c f b -a b d e +b^{2} c e}\) | \(162\) |
parallelrisch | \(\frac {x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} d^{2} f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} d^{2} f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} c d \,f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} d^{2} e f +x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} c d \,f^{2}+x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} d^{2} e f +x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} c d e f +\left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} c d e f}{b d f \left (a^{2} d f m -a b c f m -a b d e m +b^{2} c e m +a^{2} d f -a c f b -a b d e +b^{2} c e \right )}\) | \(934\) |
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (102) = 204\).
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.23 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {{\left (b^{2} d f x^{3} + a b c e + {\left (b^{2} d e + {\left (b^{2} c + a b d\right )} f\right )} x^{2} + {\left (a b c f + {\left (b^{2} c + a b d\right )} e\right )} x\right )} {\left (b x + a\right )}^{m}}{{\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f + {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} m\right )} {\left (d x + c\right )}^{\frac {2 \, b d e - {\left (b c + a d\right )} f + {\left (b d e - a d f\right )} m}{b d e - b c f}} {\left (f x + e\right )}^{\frac {b d e - {\left (b c - a d\right )} f m - {\left (2 \, b c - a d\right )} f}{b d e - b c f}}} \]
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\[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\int \left (a + b x\right )^{m} \left (c + d x\right )^{-1 - \frac {d \left (m + 1\right ) \left (- a f + b e\right )}{b \left (- c f + d e\right )}} \left (e + f x\right )^{-1 + \frac {f \left (m + 1\right ) \left (- a d + b c\right )}{b \left (- c f + d e\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (102) = 204\).
Time = 0.34 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.28 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {{\left (b^{2} x + a b\right )} e^{\left (\frac {a d f m \log \left (d x + c\right )}{b d e - b c f} - \frac {a d f m \log \left (f x + e\right )}{b d e - b c f} + \frac {a d f \log \left (d x + c\right )}{b d e - b c f} - \frac {d e m \log \left (d x + c\right )}{d e - c f} - \frac {a d f \log \left (f x + e\right )}{b d e - b c f} + \frac {c f m \log \left (f x + e\right )}{d e - c f} + m \log \left (b x + a\right ) - \frac {d e \log \left (d x + c\right )}{d e - c f} + \frac {c f \log \left (f x + e\right )}{d e - c f}\right )}}{b^{2} c e {\left (m + 1\right )} + a^{2} d f {\left (m + 1\right )} - {\left (d e {\left (m + 1\right )} + c f {\left (m + 1\right )}\right )} a b} \]
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\[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-\frac {{\left (b e - a f\right )} d {\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1} {\left (f x + e\right )}^{\frac {{\left (b c - a d\right )} f {\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1} \,d x } \]
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Time = 4.57 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.50 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {\frac {x^2\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m\,\left (b^2\,c\,f+b^2\,d\,e+a\,b\,d\,f\right )}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {b\,x\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {b^2\,d\,f\,x^3\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {a\,b\,c\,e\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}}{{\left (c+d\,x\right )}^{\frac {d\,\left (a\,f-b\,e\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}+1}} \]
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