Optimal. Leaf size=54 \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]
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Rubi [A] time = 0.0481677, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1802} \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]
Antiderivative was successfully verified.
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Rule 1802
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx &=\int \left (A b \left (1+\frac{a C}{A b}\right )+\frac{a A}{x^2}+\frac{a B}{x}+(b B+a D) x+b C x^2+b D x^3\right ) \, dx\\ &=-\frac{a A}{x}+(A b+a C) x+\frac{1}{2} (b B+a D) x^2+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4+a B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0326805, size = 54, normalized size = 1. \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 50, normalized size = 0.9 \begin{align*}{\frac{bD{x}^{4}}{4}}+{\frac{bC{x}^{3}}{3}}+{\frac{B{x}^{2}b}{2}}+{\frac{D{x}^{2}a}{2}}+Abx+aCx+aB\ln \left ( x \right ) -{\frac{Aa}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21773, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{4} \, D b x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{2} \,{\left (D a + B b\right )} x^{2} + B a \log \left (x\right ) +{\left (C a + A b\right )} x - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.302393, size = 49, normalized size = 0.91 \begin{align*} - \frac{A a}{x} + B a \log{\left (x \right )} + \frac{C b x^{3}}{3} + \frac{D b x^{4}}{4} + x^{2} \left (\frac{B b}{2} + \frac{D a}{2}\right ) + x \left (A b + C a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17958, size = 68, normalized size = 1.26 \begin{align*} \frac{1}{4} \, D b x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{2} \, D a x^{2} + \frac{1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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