Optimal. Leaf size=70 \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]
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Rubi [A] time = 0.0420841, antiderivative size = 70, 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} \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \, dx &=\int \left (a c^3+c^2 (b c+3 a d) x^2+3 c d (b c+a d) x^4+d^2 (3 b c+a d) x^6+b d^3 x^8\right ) \, dx\\ &=a c^3 x+\frac{1}{3} c^2 (b c+3 a d) x^3+\frac{3}{5} c d (b c+a d) x^5+\frac{1}{7} d^2 (3 b c+a d) x^7+\frac{1}{9} b d^3 x^9\\ \end{align*}
Mathematica [A] time = 0.0139783, size = 70, normalized size = 1. \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 73, normalized size = 1. \begin{align*}{\frac{b{d}^{3}{x}^{9}}{9}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){x}^{3}}{3}}+a{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961674, size = 95, normalized size = 1.36 \begin{align*} \frac{1}{9} \, b d^{3} x^{9} + \frac{1}{7} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{7} + \frac{3}{5} \,{\left (b c^{2} d + a c d^{2}\right )} x^{5} + a c^{3} x + \frac{1}{3} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55559, size = 169, normalized size = 2.41 \begin{align*} \frac{1}{9} x^{9} d^{3} b + \frac{3}{7} x^{7} d^{2} c b + \frac{1}{7} x^{7} d^{3} a + \frac{3}{5} x^{5} d c^{2} b + \frac{3}{5} x^{5} d^{2} c a + \frac{1}{3} x^{3} c^{3} b + x^{3} d c^{2} a + x c^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.068978, size = 76, normalized size = 1.09 \begin{align*} a c^{3} x + \frac{b d^{3} x^{9}}{9} + x^{7} \left (\frac{a d^{3}}{7} + \frac{3 b c d^{2}}{7}\right ) + x^{5} \left (\frac{3 a c d^{2}}{5} + \frac{3 b c^{2} d}{5}\right ) + x^{3} \left (a c^{2} d + \frac{b c^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14304, size = 99, normalized size = 1.41 \begin{align*} \frac{1}{9} \, b d^{3} x^{9} + \frac{3}{7} \, b c d^{2} x^{7} + \frac{1}{7} \, a d^{3} x^{7} + \frac{3}{5} \, b c^{2} d x^{5} + \frac{3}{5} \, a c d^{2} x^{5} + \frac{1}{3} \, b c^{3} x^{3} + a c^{2} d x^{3} + a c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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