Optimal. Leaf size=52 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]
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Rubi [A] time = 0.025836, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{x^2} \, dx &=\int \left (2 a A c+\frac{a^2 A}{x^2}+\frac{a^2 B}{x}+2 a B c x+A c^2 x^2+B c^2 x^3\right ) \, dx\\ &=-\frac{a^2 A}{x}+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4+a^2 B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0066716, size = 52, normalized size = 1. \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 49, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{x}}+2\,aAcx+aBc{x}^{2}+{\frac{A{c}^{2}{x}^{3}}{3}}+{\frac{B{c}^{2}{x}^{4}}{4}}+{a}^{2}B\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06795, size = 65, normalized size = 1.25 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2} \log \left (x\right ) - \frac{A a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59884, size = 131, normalized size = 2.52 \begin{align*} \frac{3 \, B c^{2} x^{5} + 4 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 24 \, A a c x^{2} + 12 \, B a^{2} x \log \left (x\right ) - 12 \, A a^{2}}{12 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.316253, size = 51, normalized size = 0.98 \begin{align*} - \frac{A a^{2}}{x} + 2 A a c x + \frac{A c^{2} x^{3}}{3} + B a^{2} \log{\left (x \right )} + B a c x^{2} + \frac{B c^{2} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13244, size = 66, normalized size = 1.27 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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