Optimal. Leaf size=37 \[ \frac {\left (a+b x+c x^2\right )^{1+m} \left (d+e x+f x^2+g x^3\right )^{1+n}}{x^2} \]
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Rubi [F]
time = 2.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\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+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5\right )}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\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+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5\right )}{x^3} \, dx &=\int \left (c e \left (1+\frac {c e (2 m+n)+b f (1+m+2 n)+a (g+3 g n)}{c e}\right ) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n-\frac {2 a d \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^3}+\frac {(-b d (1-m)-a e (1-n)) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^2}+\frac {(2 c d m+2 a f n+b e (m+n)) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x}+(2 c f (1+m+n)+b g (2+m+3 n)) x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n+c g (3+2 m+3 n) x^2 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n\right ) \, dx\\ &=-\left ((2 a d) \int \frac {\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^3} \, dx\right )+(-b d (1-m)-a e (1-n)) \int \frac {\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^2} \, dx+(c g (3+2 m+3 n)) \int x^2 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(2 c d m+2 a f n+b e (m+n)) \int \frac {\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x} \, dx+(c e (1+2 m+n)+b f (1+m+2 n)+a g (1+3 n)) \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(2 c f (1+m+n)+b g (2+m+3 n)) \int x \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx\\ \end {align*}
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Mathematica [A]
time = 9.51, size = 34, normalized size = 0.92 \begin {gather*} \frac {(a+x (b+c x))^{1+m} (d+x (e+x (f+g x)))^{1+n}}{x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 38, normalized size = 1.03
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b x +a \right )^{1+m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{1+n}}{x^{2}}\) | \(38\) |
risch | \(\frac {\left (c g \,x^{5}+b g \,x^{4}+c f \,x^{4}+a g \,x^{3}+b f \,x^{3}+c e \,x^{3}+a f \,x^{2}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) \left (c \,x^{2}+b x +a \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right )^{n}}{x^{2}}\) | \(100\) |
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. 99 vs.
\(2 (38) = 76\).
time = 0.39, size = 99, normalized size = 2.68 \begin {gather*} \frac {{\left (c g x^{5} + {\left (c f + b g\right )} x^{4} + {\left (b f + a g + c e\right )} x^{3} + {\left (c d + a f + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\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 )}}{x^{2}} \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 [B]
time = 9.22, size = 146, normalized size = 3.95 \begin {gather*} {\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n\,\left (a\,f+b\,e+c\,d+c\,g\,x^3+a\,g\,x+b\,f\,x+c\,e\,x+b\,g\,x^2+c\,f\,x^2\right )+\frac {\left (a\,e+b\,d\right )\,{\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n}{x}+\frac {a\,d\,{\left (c\,x^2+b\,x+a\right )}^m\,{\left (g\,x^3+f\,x^2+e\,x+d\right )}^n}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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