Optimal. Leaf size=56 \[ \frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7 \]
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Rubi [A] time = 0.0370527, antiderivative size = 56, 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 = {521} \[ \frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7 \]
Antiderivative was successfully verified.
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Rule 521
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx &=\int \left (a c e+(b c e+a d e+a c f) x^2+(b d e+b c f+a d f) x^4+b d f x^6\right ) \, dx\\ &=a c e x+\frac{1}{3} (b c e+a d e+a c f) x^3+\frac{1}{5} (b d e+b c f+a d f) x^5+\frac{1}{7} b d f x^7\\ \end{align*}
Mathematica [A] time = 0.0124191, size = 56, normalized size = 1. \[ \frac{1}{5} x^5 (a d f+b c f+b d e)+\frac{1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac{1}{7} b d f x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 53, normalized size = 1. \begin{align*}{\frac{bdf{x}^{7}}{7}}+{\frac{ \left ( \left ( ad+bc \right ) f+bde \right ){x}^{5}}{5}}+{\frac{ \left ( acf+ \left ( ad+bc \right ) e \right ){x}^{3}}{3}}+acex \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02574, size = 70, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b d f x^{7} + \frac{1}{5} \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{5} + a c e x + \frac{1}{3} \,{\left (a c f +{\left (b c + a d\right )} e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22783, size = 163, normalized size = 2.91 \begin{align*} \frac{1}{7} x^{7} f d b + \frac{1}{5} x^{5} e d b + \frac{1}{5} x^{5} f c b + \frac{1}{5} x^{5} f d a + \frac{1}{3} x^{3} e c b + \frac{1}{3} x^{3} e d a + \frac{1}{3} x^{3} f c a + x e c a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062564, size = 63, normalized size = 1.12 \begin{align*} a c e x + \frac{b d f x^{7}}{7} + x^{5} \left (\frac{a d f}{5} + \frac{b c f}{5} + \frac{b d e}{5}\right ) + x^{3} \left (\frac{a c f}{3} + \frac{a d e}{3} + \frac{b c e}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14974, size = 89, normalized size = 1.59 \begin{align*} \frac{1}{7} \, b d f x^{7} + \frac{1}{5} \, b c f x^{5} + \frac{1}{5} \, a d f x^{5} + \frac{1}{5} \, b d x^{5} e + \frac{1}{3} \, a c f x^{3} + \frac{1}{3} \, b c x^{3} e + \frac{1}{3} \, a d x^{3} e + a c x e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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