Integrand size = 44, antiderivative size = 246 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {2 (c d f-a e g) \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac {2 g (c d f-a e g) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac {(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)} \]
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Time = 0.14 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {884, 808, 662} \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac {2 g (d+e x)^m (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac {(f+g x)^2 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (3-m)} \]
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Rule 662
Rule 808
Rule 884
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)}+\frac {(2 (c d f-a e g)) \int (d+e x)^m (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c d (3-m)} \\ & = \frac {2 g (c d f-a e g) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac {(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)}-\frac {\left (2 (c d f-a e g) \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )\right ) \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c^2 d^2 e (2-m) (3-m)} \\ & = -\frac {2 (c d f-a e g) \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac {2 g (c d f-a e g) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac {(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.53 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {(d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \left (2 a^2 e^2 g^2+2 a c d e g (f (-3+m)+g (-1+m) x)+c^2 d^2 \left (f^2 \left (6-5 m+m^2\right )+2 f g \left (3-4 m+m^2\right ) x+g^2 \left (2-3 m+m^2\right ) x^2\right )\right )}{c^3 d^3 (-3+m) (-2+m) (-1+m)} \]
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Time = 2.12 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(-\frac {\left (c d x +a e \right ) \left (c^{2} d^{2} g^{2} m^{2} x^{2}+2 c^{2} d^{2} f g \,m^{2} x -3 c^{2} d^{2} g^{2} m \,x^{2}+2 a c d e \,g^{2} m x +c^{2} d^{2} f^{2} m^{2}-8 c^{2} d^{2} f g m x +2 g^{2} x^{2} c^{2} d^{2}+2 a c d e f g m -2 a c d e \,g^{2} x -5 c^{2} d^{2} f^{2} m +6 c^{2} d^{2} f g x +2 a^{2} e^{2} g^{2}-6 a c d e f g +6 c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c^{3} d^{3} \left (m^{3}-6 m^{2}+11 m -6\right )}\) | \(235\) |
risch | \(-\frac {\left (c^{3} d^{3} g^{2} m^{2} x^{3}+a \,c^{2} d^{2} e \,g^{2} m^{2} x^{2}+2 c^{3} d^{3} f g \,m^{2} x^{2}-3 c^{3} d^{3} g^{2} m \,x^{3}+2 a \,c^{2} d^{2} e f g \,m^{2} x -a \,c^{2} d^{2} e \,g^{2} m \,x^{2}+c^{3} d^{3} f^{2} m^{2} x -8 c^{3} d^{3} f g m \,x^{2}+2 g^{2} x^{3} c^{3} d^{3}+2 a^{2} c d \,e^{2} g^{2} m x +a \,c^{2} d^{2} e \,f^{2} m^{2}-6 a \,c^{2} d^{2} e f g m x -5 c^{3} d^{3} f^{2} m x +6 c^{3} d^{3} f g \,x^{2}+2 a^{2} c d \,e^{2} f g m -5 a \,c^{2} d^{2} e \,f^{2} m +6 c^{3} d^{3} f^{2} x +2 a^{3} e^{3} g^{2}-6 a^{2} c d \,e^{2} f g +6 a \,c^{2} d^{2} e \,f^{2}\right ) \left (c d x +a e \right )^{-m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right ) m \left (-\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (c d x +a e \right )\right )\right ) \left (-\operatorname {csgn}\left (i \left (c d x +a e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )\right )\right )}{2}}}{\left (-2+m \right ) \left (-3+m \right ) \left (-1+m \right ) c^{3} d^{3}}\) | \(406\) |
parallelrisch | \(\frac {\left (-x^{3} \left (e x +d \right )^{m} c^{3} d^{3} e \,g^{2} m^{2}+3 x^{3} \left (e x +d \right )^{m} c^{3} d^{3} e \,g^{2} m -x \left (e x +d \right )^{m} c^{3} d^{3} e \,f^{2} m^{2}-6 x^{2} \left (e x +d \right )^{m} c^{3} d^{3} e f g +5 x \left (e x +d \right )^{m} c^{3} d^{3} e \,f^{2} m -\left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} f^{2} m^{2}+5 \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} f^{2} m +6 \left (e x +d \right )^{m} a^{2} c d \,e^{3} f g -2 x \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} f g \,m^{2}+6 x \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} f g m -2 \left (e x +d \right )^{m} a^{3} e^{4} g^{2}-2 x^{3} \left (e x +d \right )^{m} c^{3} d^{3} e \,g^{2}-6 x \left (e x +d \right )^{m} c^{3} d^{3} e \,f^{2}-6 \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} f^{2}-x^{2} \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} g^{2} m^{2}-2 x^{2} \left (e x +d \right )^{m} c^{3} d^{3} e f g \,m^{2}+x^{2} \left (e x +d \right )^{m} a \,c^{2} d^{2} e^{2} g^{2} m +8 x^{2} \left (e x +d \right )^{m} c^{3} d^{3} e f g m -2 x \left (e x +d \right )^{m} a^{2} c d \,e^{3} g^{2} m -2 \left (e x +d \right )^{m} a^{2} c d \,e^{3} f g m \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{e \left (-2+m \right ) \left (-3+m \right ) \left (-1+m \right ) c^{3} d^{3}}\) | \(506\) |
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Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.42 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {{\left (a c^{2} d^{2} e f^{2} m^{2} + 6 \, a c^{2} d^{2} e f^{2} - 6 \, a^{2} c d e^{2} f g + 2 \, a^{3} e^{3} g^{2} + {\left (c^{3} d^{3} g^{2} m^{2} - 3 \, c^{3} d^{3} g^{2} m + 2 \, c^{3} d^{3} g^{2}\right )} x^{3} + {\left (6 \, c^{3} d^{3} f g + {\left (2 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} m^{2} - {\left (8 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} m\right )} x^{2} - {\left (5 \, a c^{2} d^{2} e f^{2} - 2 \, a^{2} c d e^{2} f g\right )} m + {\left (6 \, c^{3} d^{3} f^{2} + {\left (c^{3} d^{3} f^{2} + 2 \, a c^{2} d^{2} e f g\right )} m^{2} - {\left (5 \, c^{3} d^{3} f^{2} + 6 \, a c^{2} d^{2} e f g - 2 \, a^{2} c d e^{2} g^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (c^{3} d^{3} m^{3} - 6 \, c^{3} d^{3} m^{2} + 11 \, c^{3} d^{3} m - 6 \, c^{3} d^{3}\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]
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Timed out. \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.78 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {{\left (c d x + a e\right )} f^{2}}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} - \frac {2 \, {\left (c^{2} d^{2} {\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} f g}{{\left (m^{2} - 3 \, m + 2\right )} {\left (c d x + a e\right )}^{m} c^{2} d^{2}} - \frac {{\left ({\left (m^{2} - 3 \, m + 2\right )} c^{3} d^{3} x^{3} + {\left (m^{2} - m\right )} a c^{2} d^{2} e x^{2} + 2 \, a^{2} c d e^{2} m x + 2 \, a^{3} e^{3}\right )} g^{2}}{{\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} {\left (c d x + a e\right )}^{m} c^{3} d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (232) = 464\).
Time = 0.54 (sec) , antiderivative size = 929, normalized size of antiderivative = 3.78 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {{\left (e x + d\right )}^{m} c^{3} d^{3} g^{2} m^{2} x^{3} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} c^{3} d^{3} f g m^{2} x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + {\left (e x + d\right )}^{m} a c^{2} d^{2} e g^{2} m^{2} x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 3 \, {\left (e x + d\right )}^{m} c^{3} d^{3} g^{2} m x^{3} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + {\left (e x + d\right )}^{m} c^{3} d^{3} f^{2} m^{2} x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} a c^{2} d^{2} e f g m^{2} x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 8 \, {\left (e x + d\right )}^{m} c^{3} d^{3} f g m x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - {\left (e x + d\right )}^{m} a c^{2} d^{2} e g^{2} m x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} c^{3} d^{3} g^{2} x^{3} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + {\left (e x + d\right )}^{m} a c^{2} d^{2} e f^{2} m^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 5 \, {\left (e x + d\right )}^{m} c^{3} d^{3} f^{2} m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 6 \, {\left (e x + d\right )}^{m} a c^{2} d^{2} e f g m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} a^{2} c d e^{2} g^{2} m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{3} f g x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 5 \, {\left (e x + d\right )}^{m} a c^{2} d^{2} e f^{2} m e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} a^{2} c d e^{2} f g m e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{3} f^{2} x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 6 \, {\left (e x + d\right )}^{m} a c^{2} d^{2} e f^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} - 6 \, {\left (e x + d\right )}^{m} a^{2} c d e^{2} f g e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )} + 2 \, {\left (e x + d\right )}^{m} a^{3} e^{3} g^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )}}{c^{3} d^{3} m^{3} - 6 \, c^{3} d^{3} m^{2} + 11 \, c^{3} d^{3} m - 6 \, c^{3} d^{3}} \]
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Time = 12.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.33 \[ \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=-\frac {\frac {g^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2-3\,m+2\right )}{m^3-6\,m^2+11\,m-6}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (2\,a^2\,c\,d\,e^2\,g^2\,m+2\,a\,c^2\,d^2\,e\,f\,g\,m^2-6\,a\,c^2\,d^2\,e\,f\,g\,m+c^3\,d^3\,f^2\,m^2-5\,c^3\,d^3\,f^2\,m+6\,c^3\,d^3\,f^2\right )}{c^3\,d^3\,\left (m^3-6\,m^2+11\,m-6\right )}+\frac {a\,e\,{\left (d+e\,x\right )}^m\,\left (2\,a^2\,e^2\,g^2+2\,a\,c\,d\,e\,f\,g\,m-6\,a\,c\,d\,e\,f\,g+c^2\,d^2\,f^2\,m^2-5\,c^2\,d^2\,f^2\,m+6\,c^2\,d^2\,f^2\right )}{c^3\,d^3\,\left (m^3-6\,m^2+11\,m-6\right )}+\frac {g\,x^2\,\left (m-1\right )\,{\left (d+e\,x\right )}^m\,\left (a\,e\,g\,m-6\,c\,d\,f+2\,c\,d\,f\,m\right )}{c\,d\,\left (m^3-6\,m^2+11\,m-6\right )}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \]
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