Optimal. Leaf size=88 \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]
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Rubi [A] time = 0.0733669, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1671} \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]
Antiderivative was successfully verified.
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Rule 1671
Rubi steps
\begin{align*} \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d+a e x+(b d+a f) x^2+(b e+a g) x^3+(c d+b f) x^4+(c e+b g) x^5+c f x^6+c g x^7\right ) \, dx\\ &=a d x+\frac{1}{2} a e x^2+\frac{1}{3} (b d+a f) x^3+\frac{1}{4} (b e+a g) x^4+\frac{1}{5} (c d+b f) x^5+\frac{1}{6} (c e+b g) x^6+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8\\ \end{align*}
Mathematica [A] time = 0.0192433, size = 88, normalized size = 1. \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 75, normalized size = 0.9 \begin{align*} adx+{\frac{ae{x}^{2}}{2}}+{\frac{ \left ( af+bd \right ){x}^{3}}{3}}+{\frac{ \left ( ag+be \right ){x}^{4}}{4}}+{\frac{ \left ( bf+cd \right ){x}^{5}}{5}}+{\frac{ \left ( bg+ce \right ){x}^{6}}{6}}+{\frac{cf{x}^{7}}{7}}+{\frac{cg{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978737, size = 100, normalized size = 1.14 \begin{align*} \frac{1}{8} \, c g x^{8} + \frac{1}{7} \, c f x^{7} + \frac{1}{6} \,{\left (c e + b g\right )} x^{6} + \frac{1}{5} \,{\left (c d + b f\right )} x^{5} + \frac{1}{4} \,{\left (b e + a g\right )} x^{4} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b d + a f\right )} x^{3} + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53036, size = 217, normalized size = 2.47 \begin{align*} \frac{1}{8} x^{8} g c + \frac{1}{7} x^{7} f c + \frac{1}{6} x^{6} e c + \frac{1}{6} x^{6} g b + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} f b + \frac{1}{4} x^{4} e b + \frac{1}{4} x^{4} g a + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} f a + \frac{1}{2} x^{2} e a + x d a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072304, size = 83, normalized size = 0.94 \begin{align*} a d x + \frac{a e x^{2}}{2} + \frac{c f x^{7}}{7} + \frac{c g x^{8}}{8} + x^{6} \left (\frac{b g}{6} + \frac{c e}{6}\right ) + x^{5} \left (\frac{b f}{5} + \frac{c d}{5}\right ) + x^{4} \left (\frac{a g}{4} + \frac{b e}{4}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b d}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07714, size = 115, normalized size = 1.31 \begin{align*} \frac{1}{8} \, c g x^{8} + \frac{1}{7} \, c f x^{7} + \frac{1}{6} \, b g x^{6} + \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b f x^{5} + \frac{1}{4} \, a g x^{4} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a f x^{3} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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