Optimal. Leaf size=73 \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
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Rubi [A] time = 0.044485, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1153} \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d^2+d (b d+2 a e) x^2+\left (c d^2+e (2 b d+a e)\right ) x^4+e (2 c d+b e) x^6+c e^2 x^8\right ) \, dx\\ &=a d^2 x+\frac{1}{3} d (b d+2 a e) x^3+\frac{1}{5} \left (c d^2+e (2 b d+a e)\right ) x^5+\frac{1}{7} e (2 c d+b e) x^7+\frac{1}{9} c e^2 x^9\\ \end{align*}
Mathematica [A] time = 0.0161226, size = 73, normalized size = 1. \[ \frac{1}{5} x^5 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 70, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{9}}{9}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+2\,deb+c{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{3}}{3}}+a{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946527, size = 93, normalized size = 1.27 \begin{align*} \frac{1}{9} \, c e^{2} x^{9} + \frac{1}{7} \,{\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac{1}{5} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac{1}{3} \,{\left (b d^{2} + 2 \, a d e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52848, size = 185, normalized size = 2.53 \begin{align*} \frac{1}{9} x^{9} e^{2} c + \frac{2}{7} x^{7} e d c + \frac{1}{7} x^{7} e^{2} b + \frac{1}{5} x^{5} d^{2} c + \frac{2}{5} x^{5} e d b + \frac{1}{5} x^{5} e^{2} a + \frac{1}{3} x^{3} d^{2} b + \frac{2}{3} x^{3} e d a + x d^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.07416, size = 78, normalized size = 1.07 \begin{align*} a d^{2} x + \frac{c e^{2} x^{9}}{9} + x^{7} \left (\frac{b e^{2}}{7} + \frac{2 c d e}{7}\right ) + x^{5} \left (\frac{a e^{2}}{5} + \frac{2 b d e}{5} + \frac{c d^{2}}{5}\right ) + x^{3} \left (\frac{2 a d e}{3} + \frac{b d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11682, size = 103, normalized size = 1.41 \begin{align*} \frac{1}{9} \, c x^{9} e^{2} + \frac{2}{7} \, c d x^{7} e + \frac{1}{7} \, b x^{7} e^{2} + \frac{1}{5} \, c d^{2} x^{5} + \frac{2}{5} \, b d x^{5} e + \frac{1}{5} \, a x^{5} e^{2} + \frac{1}{3} \, b d^{2} x^{3} + \frac{2}{3} \, a d x^{3} e + a d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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