Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]
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Rubi [A] time = 0.0242327, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1593, 448} \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]
Antiderivative was successfully verified.
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Rule 1593
Rule 448
Rubi steps
\begin{align*} \int \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^2 \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^2+(b B+A c) x^4+B c x^6\right ) \, dx\\ &=\frac{1}{3} A b x^3+\frac{1}{5} (b B+A c) x^5+\frac{1}{7} B c x^7\\ \end{align*}
Mathematica [A] time = 0.0053023, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 28, normalized size = 0.9 \begin{align*}{\frac{Ab{x}^{3}}{3}}+{\frac{ \left ( Ac+Bb \right ){x}^{5}}{5}}+{\frac{Bc{x}^{7}}{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13554, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{5} \,{\left (B b + A c\right )} x^{5} + \frac{1}{3} \, A b x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.395012, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{7} x^{7} c B + \frac{1}{5} x^{5} b B + \frac{1}{5} x^{5} c A + \frac{1}{3} x^{3} b A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.057483, size = 29, normalized size = 0.88 \begin{align*} \frac{A b x^{3}}{3} + \frac{B c x^{7}}{7} + x^{5} \left (\frac{A c}{5} + \frac{B b}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15677, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{5} \, B b x^{5} + \frac{1}{5} \, A c x^{5} + \frac{1}{3} \, A b x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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