Optimal. Leaf size=78 \[ \frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]
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Rubi [A] time = 0.0637748, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1261} \[ \frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 1261
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d^2 x^2+d (b d+2 a e) x^4+\left (c d^2+e (2 b d+a e)\right ) x^6+e (2 c d+b e) x^8+c e^2 x^{10}\right ) \, dx\\ &=\frac{1}{3} a d^2 x^3+\frac{1}{5} d (b d+2 a e) x^5+\frac{1}{7} \left (c d^2+e (2 b d+a e)\right ) x^7+\frac{1}{9} e (2 c d+b e) x^9+\frac{1}{11} c e^2 x^{11}\\ \end{align*}
Mathematica [A] time = 0.0157542, size = 78, normalized size = 1. \[ \frac{1}{7} x^7 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 73, normalized size = 0.9 \begin{align*}{\frac{c{e}^{2}{x}^{11}}{11}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{9}}{9}}+{\frac{ \left ( a{e}^{2}+2\,deb+c{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{5}}{5}}+{\frac{a{d}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950279, size = 97, normalized size = 1.24 \begin{align*} \frac{1}{11} \, c e^{2} x^{11} + \frac{1}{9} \,{\left (2 \, c d e + b e^{2}\right )} x^{9} + \frac{1}{7} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{7} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{5} \,{\left (b d^{2} + 2 \, a d e\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48897, size = 196, normalized size = 2.51 \begin{align*} \frac{1}{11} x^{11} e^{2} c + \frac{2}{9} x^{9} e d c + \frac{1}{9} x^{9} e^{2} b + \frac{1}{7} x^{7} d^{2} c + \frac{2}{7} x^{7} e d b + \frac{1}{7} x^{7} e^{2} a + \frac{1}{5} x^{5} d^{2} b + \frac{2}{5} x^{5} e d a + \frac{1}{3} x^{3} d^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.07768, size = 82, normalized size = 1.05 \begin{align*} \frac{a d^{2} x^{3}}{3} + \frac{c e^{2} x^{11}}{11} + x^{9} \left (\frac{b e^{2}}{9} + \frac{2 c d e}{9}\right ) + x^{7} \left (\frac{a e^{2}}{7} + \frac{2 b d e}{7} + \frac{c d^{2}}{7}\right ) + x^{5} \left (\frac{2 a d e}{5} + \frac{b d^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09783, size = 107, normalized size = 1.37 \begin{align*} \frac{1}{11} \, c x^{11} e^{2} + \frac{2}{9} \, c d x^{9} e + \frac{1}{9} \, b x^{9} e^{2} + \frac{1}{7} \, c d^{2} x^{7} + \frac{2}{7} \, b d x^{7} e + \frac{1}{7} \, a x^{7} e^{2} + \frac{1}{5} \, b d^{2} x^{5} + \frac{2}{5} \, a d x^{5} e + \frac{1}{3} \, a d^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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