Optimal. Leaf size=50 \[ a^2 A x+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]
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Rubi [A] time = 0.0223412, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ a^2 A x+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A+a (2 A b+a B) x^2+b (A b+2 a B) x^4+b^2 B x^6\right ) \, dx\\ &=a^2 A x+\frac{1}{3} a (2 A b+a B) x^3+\frac{1}{5} b (A b+2 a B) x^5+\frac{1}{7} b^2 B x^7\\ \end{align*}
Mathematica [A] time = 0.0073365, size = 50, normalized size = 1. \[ a^2 A x+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{7}}{7}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{3}}{3}}+{a}^{2}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00006, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{7} \, B b^{2} x^{7} + \frac{1}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + A a^{2} x + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28013, size = 120, normalized size = 2.4 \begin{align*} \frac{1}{7} x^{7} b^{2} B + \frac{2}{5} x^{5} b a B + \frac{1}{5} x^{5} b^{2} A + \frac{1}{3} x^{3} a^{2} B + \frac{2}{3} x^{3} b a A + x a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065136, size = 53, normalized size = 1.06 \begin{align*} A a^{2} x + \frac{B b^{2} x^{7}}{7} + x^{5} \left (\frac{A b^{2}}{5} + \frac{2 B a b}{5}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{B a^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11558, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{7} \, B b^{2} x^{7} + \frac{2}{5} \, B a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{3} \, B a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + A a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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