Optimal. Leaf size=82 \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]
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Rubi [A] time = 0.0457626, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]
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 )^2 \, dx &=\int \left (a^2 c^2+2 a c (b c+a d) x^2+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+2 b d (b c+a d) x^6+b^2 d^2 x^8\right ) \, dx\\ &=a^2 c^2 x+\frac{2}{3} a c (b c+a d) x^3+\frac{1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac{2}{7} b d (b c+a d) x^7+\frac{1}{9} b^2 d^2 x^9\\ \end{align*}
Mathematica [A] time = 0.0159283, size = 82, normalized size = 1. \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 87, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,abcd+{b}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{3}}{3}}+{a}^{2}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975871, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{9} \, b^{2} d^{2} x^{9} + \frac{2}{7} \,{\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac{1}{5} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac{2}{3} \,{\left (a b c^{2} + a^{2} c d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61218, size = 209, normalized size = 2.55 \begin{align*} \frac{1}{9} x^{9} d^{2} b^{2} + \frac{2}{7} x^{7} d c b^{2} + \frac{2}{7} x^{7} d^{2} b a + \frac{1}{5} x^{5} c^{2} b^{2} + \frac{4}{5} x^{5} d c b a + \frac{1}{5} x^{5} d^{2} a^{2} + \frac{2}{3} x^{3} c^{2} b a + \frac{2}{3} x^{3} d c a^{2} + x c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072224, size = 97, normalized size = 1.18 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{9}}{9} + x^{7} \left (\frac{2 a b d^{2}}{7} + \frac{2 b^{2} c d}{7}\right ) + x^{5} \left (\frac{a^{2} d^{2}}{5} + \frac{4 a b c d}{5} + \frac{b^{2} c^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{2} c d}{3} + \frac{2 a b c^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18338, size = 123, normalized size = 1.5 \begin{align*} \frac{1}{9} \, b^{2} d^{2} x^{9} + \frac{2}{7} \, b^{2} c d x^{7} + \frac{2}{7} \, a b d^{2} x^{7} + \frac{1}{5} \, b^{2} c^{2} x^{5} + \frac{4}{5} \, a b c d x^{5} + \frac{1}{5} \, a^{2} d^{2} x^{5} + \frac{2}{3} \, a b c^{2} x^{3} + \frac{2}{3} \, a^{2} c d x^{3} + a^{2} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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