Optimal. Leaf size=48 \[ -\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3 \]
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Rubi [A] time = 0.0278166, 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} \[ -\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3 \]
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^4} \, dx &=\int \left (b (b c+2 a d)+\frac{a^2 c}{x^4}+\frac{a (2 b c+a d)}{x^2}+b^2 d x^2\right ) \, dx\\ &=-\frac{a^2 c}{3 x^3}-\frac{a (2 b c+a d)}{x}+b (b c+2 a d) x+\frac{1}{3} b^2 d x^3\\ \end{align*}
Mathematica [A] time = 0.0193895, size = 50, normalized size = 1.04 \[ \frac{a^2 (-d)-2 a b c}{x}-\frac{a^2 c}{3 x^3}+b x (2 a d+b c)+\frac{1}{3} b^2 d x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 46, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{3}}{3}}+2\,abdx+{b}^{2}cx-{\frac{{a}^{2}c}{3\,{x}^{3}}}-{\frac{a \left ( ad+2\,bc \right ) }{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07129, size = 68, normalized size = 1.42 \begin{align*} \frac{1}{3} \, b^{2} d x^{3} +{\left (b^{2} c + 2 \, a b d\right )} x - \frac{a^{2} c + 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11661, size = 109, normalized size = 2.27 \begin{align*} \frac{b^{2} d x^{6} + 3 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - a^{2} c - 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.381036, size = 49, normalized size = 1.02 \begin{align*} \frac{b^{2} d x^{3}}{3} + x \left (2 a b d + b^{2} c\right ) - \frac{a^{2} c + x^{2} \left (3 a^{2} d + 6 a b c\right )}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15255, size = 68, normalized size = 1.42 \begin{align*} \frac{1}{3} \, b^{2} d x^{3} + b^{2} c x + 2 \, a b d x - \frac{6 \, a b c x^{2} + 3 \, a^{2} d x^{2} + a^{2} c}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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