Optimal. Leaf size=28 \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]
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Rubi [A] time = 0.0128864, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {373} \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \, dx &=\int \left (a c+(b c+a d) x^2+b d x^4\right ) \, dx\\ &=a c x+\frac{1}{3} (b c+a d) x^3+\frac{1}{5} b d x^5\\ \end{align*}
Mathematica [A] time = 0.005004, size = 28, normalized size = 1. \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( ad+bc \right ){x}^{3}}{3}}+{\frac{bd{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46598, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{5} \, b d x^{5} + \frac{1}{3} \,{\left (b c + a d\right )} x^{3} + a c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51667, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{5} x^{5} d b + \frac{1}{3} x^{3} c b + \frac{1}{3} x^{3} d a + x c a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.054436, size = 26, normalized size = 0.93 \begin{align*} a c x + \frac{b d x^{5}}{5} + x^{3} \left (\frac{a d}{3} + \frac{b c}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08404, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{5} \, b d x^{5} + \frac{1}{3} \, b c x^{3} + \frac{1}{3} \, a d x^{3} + a c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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