Integrand size = 24, antiderivative size = 114 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}-\frac {(a d f (2+m)-b (d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^2 (1+m) (2+m)} \]
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Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {80, 37} \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+2)+b c f (m+1)+b d e)}{d (m+1) (m+2) (b c-a d)^2} \]
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Rule 37
Rule 80
Rubi steps \begin{align*} \text {integral}& = \frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}+\frac {(b d e+b c f (1+m)-a d f (2+m)) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d (b c-a d) (2+m)} \\ & = \frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}+\frac {(b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^2 (1+m) (2+m)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.72 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-2-m} (b (c e (2+m)+d e x+c f (1+m) x)-a (c f+d e (1+m)+d f (2+m) x))}{(b c-a d)^2 (1+m) (2+m)} \]
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Time = 1.73 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.39
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{-2-m} \left (a d f m x -b c f m x +a d e m +2 a d f x -b c e m -b c f x -b d e x +a c f +a d e -2 b c e \right )}{a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(158\) |
parallelrisch | \(-\frac {x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b \,d^{3} f m +x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} d^{3} e m -x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c^{2} d f m +x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} d^{3} f m -x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c \,d^{2} f m +3 x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b c \,d^{2} f -2 x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} c \,d^{2} e +\left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b c \,d^{2} e m -\left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} c^{2} d e m -x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c \,d^{2} e m +2 x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} c \,d^{2} f +x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b \,d^{3} e m -x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c^{2} d e m +x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b c \,d^{2} f m -x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} c^{2} d f m +2 x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} d^{3} f -x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c \,d^{2} f +2 x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b \,d^{3} f -x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c^{2} d f -3 x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c \,d^{2} e +x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b \,d^{3} e -2 x \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} c^{2} d e +\left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b \,c^{2} d f +\left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a^{2} b c \,d^{2} e -2 \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} a \,b^{2} c^{2} d e -x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} b^{3} d^{3} e}{\left (a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) b d}\) | \(876\) |
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Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (114) = 228\).
Time = 0.25 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.95 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=-\frac {{\left (a^{2} c^{2} f - {\left (b^{2} d^{2} e + {\left (b^{2} c d - a b d^{2}\right )} f m + {\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} e m - {\left (3 \, b^{2} c d e + {\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2}\right )} f + {\left ({\left (b^{2} c d - a b d^{2}\right )} e + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f\right )} m\right )} x^{2} - {\left (2 \, a b c^{2} - a^{2} c d\right )} e + {\left (3 \, a^{2} c d f - {\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2}\right )} e - {\left ({\left (b^{2} c^{2} - a^{2} d^{2}\right )} e + {\left (a b c^{2} - a^{2} c d\right )} f\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m^{2} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m} \]
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Exception generated. \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\int { {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (114) = 228\).
Time = 0.29 (sec) , antiderivative size = 1023, normalized size of antiderivative = 8.97 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\text {Too large to display} \]
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Time = 3.16 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.16 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,e-2\,b^2\,c^2\,e+3\,a^2\,c\,d\,f+a^2\,d^2\,e\,m-b^2\,c^2\,e\,m-2\,a\,b\,c\,d\,e-a\,b\,c^2\,f\,m+a^2\,c\,d\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a\,c\,f+a\,d\,e-2\,b\,c\,e+a\,d\,e\,m-b\,c\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (2\,a^2\,d^2\,f-b^2\,c^2\,f-3\,b^2\,c\,d\,e+a^2\,d^2\,f\,m-b^2\,c^2\,f\,m+2\,a\,b\,c\,d\,f+a\,b\,d^2\,e\,m-b^2\,c\,d\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )} \]
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