Optimal. Leaf size=122 \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]
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Rubi [A] time = 0.0700041, antiderivative size = 122, 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} \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^3 \left (c+d x^2\right )^2 \, dx &=\int \left (a^3 c^2+a^2 c (3 b c+2 a d) x^2+a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^6+b^2 d (2 b c+3 a d) x^8+b^3 d^2 x^{10}\right ) \, dx\\ &=a^3 c^2 x+\frac{1}{3} a^2 c (3 b c+2 a d) x^3+\frac{1}{5} a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^5+\frac{1}{7} b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac{1}{9} b^2 d (2 b c+3 a d) x^9+\frac{1}{11} b^3 d^2 x^{11}\\ \end{align*}
Mathematica [A] time = 0.020559, size = 122, normalized size = 1. \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 125, normalized size = 1. \begin{align*}{\frac{{b}^{3}{d}^{2}{x}^{11}}{11}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{2}+2\,{b}^{3}cd \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{2}+6\,a{b}^{2}cd+{b}^{3}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}{d}^{2}+6\,{a}^{2}bcd+3\,a{b}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3}cd+3\,{a}^{2}b{c}^{2} \right ){x}^{3}}{3}}+{a}^{3}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982753, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{1}{9} \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac{1}{5} \,{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47619, size = 296, normalized size = 2.43 \begin{align*} \frac{1}{11} x^{11} d^{2} b^{3} + \frac{2}{9} x^{9} d c b^{3} + \frac{1}{3} x^{9} d^{2} b^{2} a + \frac{1}{7} x^{7} c^{2} b^{3} + \frac{6}{7} x^{7} d c b^{2} a + \frac{3}{7} x^{7} d^{2} b a^{2} + \frac{3}{5} x^{5} c^{2} b^{2} a + \frac{6}{5} x^{5} d c b a^{2} + \frac{1}{5} x^{5} d^{2} a^{3} + x^{3} c^{2} b a^{2} + \frac{2}{3} x^{3} d c a^{3} + x c^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080502, size = 136, normalized size = 1.11 \begin{align*} a^{3} c^{2} x + \frac{b^{3} d^{2} x^{11}}{11} + x^{9} \left (\frac{a b^{2} d^{2}}{3} + \frac{2 b^{3} c d}{9}\right ) + x^{7} \left (\frac{3 a^{2} b d^{2}}{7} + \frac{6 a b^{2} c d}{7} + \frac{b^{3} c^{2}}{7}\right ) + x^{5} \left (\frac{a^{3} d^{2}}{5} + \frac{6 a^{2} b c d}{5} + \frac{3 a b^{2} c^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{3} c d}{3} + a^{2} b c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08105, size = 177, normalized size = 1.45 \begin{align*} \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{2}{9} \, b^{3} c d x^{9} + \frac{1}{3} \, a b^{2} d^{2} x^{9} + \frac{1}{7} \, b^{3} c^{2} x^{7} + \frac{6}{7} \, a b^{2} c d x^{7} + \frac{3}{7} \, a^{2} b d^{2} x^{7} + \frac{3}{5} \, a b^{2} c^{2} x^{5} + \frac{6}{5} \, a^{2} b c d x^{5} + \frac{1}{5} \, a^{3} d^{2} x^{5} + a^{2} b c^{2} x^{3} + \frac{2}{3} \, a^{3} c d x^{3} + a^{3} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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