Optimal. Leaf size=54 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]
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Rubi [A] time = 0.048458, 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} \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]
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^3} \, dx &=\int \left (b B \left (1+\frac{a D}{b B}\right )+\frac{a A}{x^3}+\frac{a B}{x^2}+\frac{A b+a C}{x}+b C x+b D x^2\right ) \, dx\\ &=-\frac{a A}{2 x^2}-\frac{a B}{x}+(b B+a D) x+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3+(A b+a C) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0261517, size = 51, normalized size = 0.94 \[ \log (x) (a C+A b)-\frac{a \left (A+2 B x-2 D x^3\right )}{2 x^2}+\frac{1}{6} b x \left (6 B+3 C x+2 D x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 48, normalized size = 0.9 \begin{align*}{\frac{bD{x}^{3}}{3}}+{\frac{bC{x}^{2}}{2}}+bBx+aDx+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ba}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06238, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} +{\left (D a + B b\right )} x +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \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 = 0.415662, size = 49, normalized size = 0.91 \begin{align*} \frac{C b x^{2}}{2} + \frac{D b x^{3}}{3} + x \left (B b + D a\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15191, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} + D a x + B b x +{\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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