Optimal. Leaf size=81 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]
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Rubi [A] time = 0.0426554, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx &=\int \left (2 a c (b c+a d)+\frac{a^2 c^2}{x^2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^2+2 b d (b c+a d) x^4+b^2 d^2 x^6\right ) \, dx\\ &=-\frac{a^2 c^2}{x}+2 a c (b c+a d) x+\frac{1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac{2}{5} b d (b c+a d) x^5+\frac{1}{7} b^2 d^2 x^7\\ \end{align*}
Mathematica [A] time = 0.0377618, size = 81, normalized size = 1. \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{x}+\frac{2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac{1}{7} b^2 d^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 91, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{7}}{7}}+{\frac{2\,{x}^{5}ab{d}^{2}}{5}}+{\frac{2\,{x}^{5}{b}^{2}cd}{5}}+{\frac{{x}^{3}{a}^{2}{d}^{2}}{3}}+{\frac{4\,{x}^{3}abcd}{3}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3}}+2\,{a}^{2}cdx+2\,ab{c}^{2}x-{\frac{{a}^{2}{c}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00465, size = 112, normalized size = 1.38 \begin{align*} \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \,{\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac{a^{2} c^{2}}{x} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28043, size = 189, normalized size = 2.33 \begin{align*} \frac{15 \, b^{2} d^{2} x^{8} + 42 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.315012, size = 92, normalized size = 1.14 \begin{align*} - \frac{a^{2} c^{2}}{x} + \frac{b^{2} d^{2} x^{7}}{7} + x^{5} \left (\frac{2 a b d^{2}}{5} + \frac{2 b^{2} c d}{5}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3286, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{7} \, b^{2} d^{2} x^{7} + \frac{2}{5} \, b^{2} c d x^{5} + \frac{2}{5} \, a b d^{2} x^{5} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac{a^{2} c^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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