Optimal. Leaf size=120 \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]
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Rubi [A] time = 0.0564373, antiderivative size = 120, 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} \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]
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 )^3}{x^2} \, dx &=\int \left (a c^2 (2 b c+3 a d)+\frac{a^2 c^3}{x^2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^2+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+b d^2 (3 b c+2 a d) x^6+b^2 d^3 x^8\right ) \, dx\\ &=-\frac{a^2 c^3}{x}+a c^2 (2 b c+3 a d) x+\frac{1}{3} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^3+\frac{1}{5} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^5+\frac{1}{7} b d^2 (3 b c+2 a d) x^7+\frac{1}{9} b^2 d^3 x^9\\ \end{align*}
Mathematica [A] time = 0.0381116, size = 120, normalized size = 1. \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 131, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{9}}{9}}+{\frac{2\,{x}^{7}ab{d}^{3}}{7}}+{\frac{3\,{x}^{7}{b}^{2}c{d}^{2}}{7}}+{\frac{{x}^{5}{a}^{2}{d}^{3}}{5}}+{\frac{6\,{x}^{5}abc{d}^{2}}{5}}+{\frac{3\,{x}^{5}{b}^{2}{c}^{2}d}{5}}+{x}^{3}{a}^{2}c{d}^{2}+2\,{x}^{3}ab{c}^{2}d+{\frac{{x}^{3}{b}^{2}{c}^{3}}{3}}+3\,{a}^{2}{c}^{2}dx+2\,ab{c}^{3}x-{\frac{{a}^{2}{c}^{3}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991279, size = 167, normalized size = 1.39 \begin{align*} \frac{1}{9} \, b^{2} d^{3} x^{9} + \frac{1}{7} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{7} + \frac{1}{5} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{5} - \frac{a^{2} c^{3}}{x} + \frac{1}{3} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} +{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21753, size = 279, normalized size = 2.32 \begin{align*} \frac{35 \, b^{2} d^{3} x^{10} + 45 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 63 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 315 \, a^{2} c^{3} + 105 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 315 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{315 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.360903, size = 131, normalized size = 1.09 \begin{align*} - \frac{a^{2} c^{3}}{x} + \frac{b^{2} d^{3} x^{9}}{9} + x^{7} \left (\frac{2 a b d^{3}}{7} + \frac{3 b^{2} c d^{2}}{7}\right ) + x^{5} \left (\frac{a^{2} d^{3}}{5} + \frac{6 a b c d^{2}}{5} + \frac{3 b^{2} c^{2} d}{5}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac{b^{2} c^{3}}{3}\right ) + x \left (3 a^{2} c^{2} d + 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 = 1.12283, size = 176, normalized size = 1.47 \begin{align*} \frac{1}{9} \, b^{2} d^{3} x^{9} + \frac{3}{7} \, b^{2} c d^{2} x^{7} + \frac{2}{7} \, a b d^{3} x^{7} + \frac{3}{5} \, b^{2} c^{2} d x^{5} + \frac{6}{5} \, a b c d^{2} x^{5} + \frac{1}{5} \, a^{2} d^{3} x^{5} + \frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + 2 \, a b c^{3} x + 3 \, a^{2} c^{2} d x - \frac{a^{2} c^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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