Optimal. Leaf size=48 \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
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Rubi [A] time = 0.0256699, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
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 )}{x^2} \, dx &=\int \left (a (2 b c+a d)+\frac{a^2 c}{x^2}+b (b c+2 a d) x^2+b^2 d x^4\right ) \, dx\\ &=-\frac{a^2 c}{x}+a (2 b c+a d) x+\frac{1}{3} b (b c+2 a d) x^3+\frac{1}{5} b^2 d x^5\\ \end{align*}
Mathematica [A] time = 0.0161062, size = 48, normalized size = 1. \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{2\,{x}^{3}abd}{3}}+{\frac{{x}^{3}{b}^{2}c}{3}}+{a}^{2}dx+2\,abcx-{\frac{{a}^{2}c}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00418, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} - \frac{a^{2} c}{x} +{\left (2 \, a b c + a^{2} d\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26258, size = 116, normalized size = 2.42 \begin{align*} \frac{3 \, b^{2} d x^{6} + 5 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 15 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{15 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.281134, size = 48, normalized size = 1. \begin{align*} - \frac{a^{2} c}{x} + \frac{b^{2} d x^{5}}{5} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x \left (a^{2} d + 2 a b c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16103, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + 2 \, a b c x + a^{2} d x - \frac{a^{2} c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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