Optimal. Leaf size=234 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{9} x^9 \left (2 c (a h+b f)+b^2 h+c^2 d\right )+\frac{1}{7} x^7 \left (2 b (a h+c d)+2 a c f+b^2 f\right )+\frac{1}{5} x^5 \left (2 a b f+a (a h+2 c d)+b^2 d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c x^{11} (2 b h+c f)+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13} \]
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Rubi [A] time = 0.238095, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1671} \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{9} x^9 \left (2 c (a h+b f)+b^2 h+c^2 d\right )+\frac{1}{7} x^7 \left (2 b (a h+c d)+2 a c f+b^2 f\right )+\frac{1}{5} x^5 \left (2 a b f+a (a h+2 c d)+b^2 d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c x^{11} (2 b h+c f)+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13} \]
Antiderivative was successfully verified.
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Rule 1671
Rubi steps
\begin{align*} \int \left (a+b x^2+c x^4\right )^2 \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx &=\int \left (a^2 d+a^2 e x+a (2 b d+a f) x^2+a (2 b e+a g) x^3+\left (b^2 d+2 a b f+a (2 c d+a h)\right ) x^4+\left (b^2 e+2 a c e+2 a b g\right ) x^5+\left (b^2 f+2 a c f+2 b (c d+a h)\right ) x^6+\left (2 b c e+b^2 g+2 a c g\right ) x^7+\left (c^2 d+b^2 h+2 c (b f+a h)\right ) x^8+c (c e+2 b g) x^9+c (c f+2 b h) x^{10}+c^2 g x^{11}+c^2 h x^{12}\right ) \, dx\\ &=a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{3} a (2 b d+a f) x^3+\frac{1}{4} a (2 b e+a g) x^4+\frac{1}{5} \left (b^2 d+2 a b f+a (2 c d+a h)\right ) x^5+\frac{1}{6} \left (b^2 e+2 a c e+2 a b g\right ) x^6+\frac{1}{7} \left (b^2 f+2 a c f+2 b (c d+a h)\right ) x^7+\frac{1}{8} \left (2 b c e+b^2 g+2 a c g\right ) x^8+\frac{1}{9} \left (c^2 d+b^2 h+2 c (b f+a h)\right ) x^9+\frac{1}{10} c (c e+2 b g) x^{10}+\frac{1}{11} c (c f+2 b h) x^{11}+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13}\\ \end{align*}
Mathematica [A] time = 0.0877014, size = 234, normalized size = 1. \[ \frac{1}{5} x^5 \left (a^2 h+2 a b f+2 a c d+b^2 d\right )+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{9} x^9 \left (2 a c h+b^2 h+2 b c f+c^2 d\right )+\frac{1}{7} x^7 \left (2 a b h+2 a c f+b^2 f+2 b c d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c x^{11} (2 b h+c f)+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 219, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}h{x}^{13}}{13}}+{\frac{{c}^{2}g{x}^{12}}{12}}+{\frac{ \left ( 2\,bch+{c}^{2}f \right ){x}^{11}}{11}}+{\frac{ \left ( 2\,gbc+e{c}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 2\,ac+{b}^{2} \right ) h+2\,fbc+{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bce+g \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,abh+f \left ( 2\,ac+{b}^{2} \right ) +2\,bcd \right ){x}^{7}}{7}}+{\frac{ \left ( e \left ( 2\,ac+{b}^{2} \right ) +2\,abg \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}h+2\,abf+d \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( g{a}^{2}+2\,eab \right ){x}^{4}}{4}}+{\frac{ \left ( f{a}^{2}+2\,dab \right ){x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{2}}{2}}+{a}^{2}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937189, size = 294, normalized size = 1.26 \begin{align*} \frac{1}{13} \, c^{2} h x^{13} + \frac{1}{12} \, c^{2} g x^{12} + \frac{1}{11} \,{\left (c^{2} f + 2 \, b c h\right )} x^{11} + \frac{1}{10} \,{\left (c^{2} e + 2 \, b c g\right )} x^{10} + \frac{1}{9} \,{\left (c^{2} d + 2 \, b c f +{\left (b^{2} + 2 \, a c\right )} h\right )} x^{9} + \frac{1}{8} \,{\left (2 \, b c e +{\left (b^{2} + 2 \, a c\right )} g\right )} x^{8} + \frac{1}{7} \,{\left (2 \, b c d + 2 \, a b h +{\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac{1}{6} \,{\left (2 \, a b g +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (2 \, a b f + a^{2} h +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac{1}{2} \, a^{2} e x^{2} + \frac{1}{4} \,{\left (2 \, a b e + a^{2} g\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b d + a^{2} f\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73156, size = 653, normalized size = 2.79 \begin{align*} \frac{1}{13} x^{13} h c^{2} + \frac{1}{12} x^{12} g c^{2} + \frac{1}{11} x^{11} f c^{2} + \frac{2}{11} x^{11} h c b + \frac{1}{10} x^{10} e c^{2} + \frac{1}{5} x^{10} g c b + \frac{1}{9} x^{9} d c^{2} + \frac{2}{9} x^{9} f c b + \frac{1}{9} x^{9} h b^{2} + \frac{2}{9} x^{9} h c a + \frac{1}{4} x^{8} e c b + \frac{1}{8} x^{8} g b^{2} + \frac{1}{4} x^{8} g c a + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} f b^{2} + \frac{2}{7} x^{7} f c a + \frac{2}{7} x^{7} h b a + \frac{1}{6} x^{6} e b^{2} + \frac{1}{3} x^{6} e c a + \frac{1}{3} x^{6} g b a + \frac{1}{5} x^{5} d b^{2} + \frac{2}{5} x^{5} d c a + \frac{2}{5} x^{5} f b a + \frac{1}{5} x^{5} h a^{2} + \frac{1}{2} x^{4} e b a + \frac{1}{4} x^{4} g a^{2} + \frac{2}{3} x^{3} d b a + \frac{1}{3} x^{3} f a^{2} + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104479, size = 258, normalized size = 1.1 \begin{align*} a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{c^{2} g x^{12}}{12} + \frac{c^{2} h x^{13}}{13} + x^{11} \left (\frac{2 b c h}{11} + \frac{c^{2} f}{11}\right ) + x^{10} \left (\frac{b c g}{5} + \frac{c^{2} e}{10}\right ) + x^{9} \left (\frac{2 a c h}{9} + \frac{b^{2} h}{9} + \frac{2 b c f}{9} + \frac{c^{2} d}{9}\right ) + x^{8} \left (\frac{a c g}{4} + \frac{b^{2} g}{8} + \frac{b c e}{4}\right ) + x^{7} \left (\frac{2 a b h}{7} + \frac{2 a c f}{7} + \frac{b^{2} f}{7} + \frac{2 b c d}{7}\right ) + x^{6} \left (\frac{a b g}{3} + \frac{a c e}{3} + \frac{b^{2} e}{6}\right ) + x^{5} \left (\frac{a^{2} h}{5} + \frac{2 a b f}{5} + \frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) + x^{4} \left (\frac{a^{2} g}{4} + \frac{a b e}{2}\right ) + x^{3} \left (\frac{a^{2} f}{3} + \frac{2 a b d}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08647, size = 350, normalized size = 1.5 \begin{align*} \frac{1}{13} \, c^{2} h x^{13} + \frac{1}{12} \, c^{2} g x^{12} + \frac{1}{11} \, c^{2} f x^{11} + \frac{2}{11} \, b c h x^{11} + \frac{1}{5} \, b c g x^{10} + \frac{1}{10} \, c^{2} x^{10} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{9} \, b c f x^{9} + \frac{1}{9} \, b^{2} h x^{9} + \frac{2}{9} \, a c h x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{4} \, a c g x^{8} + \frac{1}{4} \, b c x^{8} e + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, b^{2} f x^{7} + \frac{2}{7} \, a c f x^{7} + \frac{2}{7} \, a b h x^{7} + \frac{1}{3} \, a b g x^{6} + \frac{1}{6} \, b^{2} x^{6} e + \frac{1}{3} \, a c x^{6} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a c d x^{5} + \frac{2}{5} \, a b f x^{5} + \frac{1}{5} \, a^{2} h x^{5} + \frac{1}{4} \, a^{2} g x^{4} + \frac{1}{2} \, a b x^{4} e + \frac{2}{3} \, a b d x^{3} + \frac{1}{3} \, a^{2} f x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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