Optimal. Leaf size=42 \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]
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Rubi [A] time = 0.0273412, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1153} \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d+(b d+a e) x^2+(c d+b e) x^4+c e x^6\right ) \, dx\\ &=a d x+\frac{1}{3} (b d+a e) x^3+\frac{1}{5} (c d+b e) x^5+\frac{1}{7} c e x^7\\ \end{align*}
Mathematica [A] time = 0.0081182, size = 42, normalized size = 1. \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 37, normalized size = 0.9 \begin{align*} adx+{\frac{ \left ( ae+bd \right ){x}^{3}}{3}}+{\frac{ \left ( be+cd \right ){x}^{5}}{5}}+{\frac{ce{x}^{7}}{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950216, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{7} \, c e x^{7} + \frac{1}{5} \,{\left (c d + b e\right )} x^{5} + \frac{1}{3} \,{\left (b d + a e\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 = 1.38201, size = 104, normalized size = 2.48 \begin{align*} \frac{1}{7} x^{7} e c + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} e b + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} e a + x d a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.063286, size = 39, normalized size = 0.93 \begin{align*} a d x + \frac{c e x^{7}}{7} + x^{5} \left (\frac{b e}{5} + \frac{c d}{5}\right ) + x^{3} \left (\frac{a e}{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 = 1.13787, size = 58, normalized size = 1.38 \begin{align*} \frac{1}{7} \, c x^{7} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b x^{5} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a x^{3} e + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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