Optimal. Leaf size=50 \[ a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7 \]
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Rubi [A] time = 0.0250865, 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} \[ a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7 \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c+a (2 b c+a d) x^2+b (b c+2 a d) x^4+b^2 d x^6\right ) \, dx\\ &=a^2 c x+\frac{1}{3} a (2 b c+a d) x^3+\frac{1}{5} b (b c+2 a d) x^5+\frac{1}{7} b^2 d x^7\\ \end{align*}
Mathematica [A] time = 0.0074746, size = 50, normalized size = 1. \[ a^2 c x+\frac{1}{5} b x^5 (2 a d+b c)+\frac{1}{3} a x^3 (a d+2 b c)+\frac{1}{7} b^2 d x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{7}}{7}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{3}}{3}}+{a}^{2}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00811, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{7} \, b^{2} d x^{7} + \frac{1}{5} \,{\left (b^{2} c + 2 \, a b d\right )} x^{5} + a^{2} c x + \frac{1}{3} \,{\left (2 \, a b c + a^{2} d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07312, size = 120, normalized size = 2.4 \begin{align*} \frac{1}{7} x^{7} d b^{2} + \frac{1}{5} x^{5} c b^{2} + \frac{2}{5} x^{5} d b a + \frac{2}{3} x^{3} c b a + \frac{1}{3} x^{3} d a^{2} + x c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067187, size = 53, normalized size = 1.06 \begin{align*} a^{2} c x + \frac{b^{2} d x^{7}}{7} + x^{5} \left (\frac{2 a b d}{5} + \frac{b^{2} c}{5}\right ) + x^{3} \left (\frac{a^{2} d}{3} + \frac{2 a b c}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14705, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{7} \, b^{2} d x^{7} + \frac{1}{5} \, b^{2} c x^{5} + \frac{2}{5} \, a b d x^{5} + \frac{2}{3} \, a b c x^{3} + \frac{1}{3} \, a^{2} d x^{3} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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