Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^{m+1} \left (d+e x+f x^2+g x^3\right )^{n+1}}{x} \]
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Rubi [F] time = 3.37432, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \left (-a d+(b d m+a e n) x+(c d+b e+a f+2 c d m+b e m+b e n+2 a f n) x^2+(2 c e+2 b f+2 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(3 c f+3 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (4+2 m+3 n) x^5\right )}{x^2} \, dx \]
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 (-a d+(b d m+a e n) x+(c d+b e+a f+2 c d m+b e m+b e n+2 a f n) x^2+(2 c e+2 b f+2 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(3 c f+3 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (4+2 m+3 n) x^5\right )}{x^2} \, dx &=\int \left (c d \left (1+\frac{2 c d m+b e (1+m+n)+a (f+2 f n)}{c d}\right ) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n-\frac{a d \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x^2}+\frac{(b d m+a e n) \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n}{x}+(c e (2+2 m+n)+b f (2+m+2 n)+a g (2+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 f (3+2 m+2 n)+b g (3+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+c g (4+2 m+3 n) x^3 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n\right ) \, dx\\ &=-\left ((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^2} \, dx\right )+(c g (4+2 m+3 n)) \int x^3 \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(b d m+a e 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 (d+2 d m)+b e (1+m+n)+a f (1+2 n)) \int \left (a+b x+c x^2\right )^m \left (d+e x+f x^2+g x^3\right )^n \, dx+(c e (2+2 m+n)+b f (2+m+2 n)+a g (2+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+(c f (3+2 m+2 n)+b g (3+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\\ \end{align*}
Mathematica [A] time = 0.950024, size = 34, normalized size = 0.92 \[ \frac{(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) ^{1+n}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48582, size = 128, normalized size = 3.46 \begin{align*} \frac{{\left (c g x^{5} +{\left (c f + b g\right )} x^{4} +{\left (c e + b f + a g\right )} x^{3} +{\left (c d + b e + a f\right )} x^{2} + a d +{\left (b d + a e\right )} x\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 )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c g{\left (2 \, m + 3 \, n + 4\right )} x^{5} +{\left (2 \, c f m + b g m + 2 \, c f n + 3 \, b g n + 3 \, c f + 3 \, b g\right )} x^{4} +{\left (2 \, c e m + b f m + c e n + 2 \, b f n + 3 \, a g n + 2 \, c e + 2 \, b f + 2 \, a g\right )} x^{3} +{\left (2 \, c d m + b e m + b e n + 2 \, a f n + c d + b e + a f\right )} x^{2} - a d +{\left (b d m + a e n\right )} x\right )}{\left (g x^{3} + f x^{2} + e x + d\right )}^{n}{\left (c x^{2} + b x + a\right )}^{m}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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