Optimal. Leaf size=37 \[ x^2 \left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n} \]
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Rubi [A]
time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 176, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {1604}
\begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1604
Rubi steps
\begin {align*} \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}\\ \end {align*}
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Mathematica [A]
time = 4.04, size = 34, normalized size = 0.92 \begin {gather*} x^2 (a+x (b+c x))^{1+m} (d+x (e+x (f+g x)))^{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 38, normalized size = 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 | \(x^{2} \left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c e \,x^{3}+a f \,x^{2}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (38) = 76\).
time = 0.39, size = 101, normalized size = 2.73 \begin {gather*} {\left (c g x^{7} + {\left (c f + b g\right )} x^{6} + {\left (b f + a g + c e\right )} x^{5} + {\left (c d + a f + b e\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} + x e + d\right ) + m \log \left (c x^{2} + b x + a\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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