Integrand size = 176, antiderivative size = 37 \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=x^2 \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n} \]
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Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {1604} \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=x^2 \left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = x^2 \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n} \\ \end{align*}
Time = 3.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=x^2 (a+x (b+c x))^{1+m} (d+x (e+x (f+g x)))^{1+n} \]
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Time = 56.81 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(x^{2} \left (c \,x^{2}+b x +a \right )^{1+m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{1+n}\) | \(38\) |
risch | \(x^{2} \left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c \,x^{3} e +a f \,x^{2}+e \,x^{2} b +x^{2} c d +a e x +b d x +a d \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n}\) | \(100\) |
parallelrisch | \(\frac {x^{7} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} g^{2}+x^{6} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c \,g^{2}+x^{6} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} f g +x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c \,g^{2}+x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c f g +x^{5} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} e g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c f g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c e g +x^{4} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} c^{2} d g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c e g +x^{3} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} b c d g +x^{2} \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n} a c d g}{g c}\) | \(460\) |
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Timed out. \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.62 \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx={\left (c g x^{7} + {\left (c f + b g\right )} x^{6} + {\left (c e + b f + a g\right )} x^{5} + {\left (c d + b e + a f\right )} x^{4} + a d x^{2} + {\left (b d + a e\right )} x^{3}\right )} e^{\left (n \log \left (g x^{3} + f x^{2} + e x + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \]
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Timed out. \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right ) \, dx=\text {Hanged} \]
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