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 = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 1.00 \[ x (a C+A b)-\frac {a A}{x}+\frac {1}{2} x^2 (a D+b B)+a B \log (x)+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 55, normalized size = 1.02 \[ \frac {3 \, D b x^{5} + 4 \, C b x^{4} + 6 \, {\left (D a + B b\right )} x^{3} + 12 \, B a x \log \relax (x) + 12 \, {\left (C a + A b\right )} x^{2} - 12 \, A a}{12 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 50, normalized size = 0.93 \[ \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, D a x^{2} + \frac {1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.93 \[ \frac {D b \,x^{4}}{4}+\frac {C b \,x^{3}}{3}+\frac {B b \,x^{2}}{2}+\frac {D a \,x^{2}}{2}+A b x +B a \ln \relax (x )+C a x -\frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 48, normalized size = 0.89 \[ \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, {\left (D a + B b\right )} x^{2} + B a \log \relax (x) + {\left (C a + A b\right )} x - \frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 49, normalized size = 0.91 \[ \frac {a\,x^2\,D}{2}+\frac {b\,x^4\,D}{4}+A\,b\,x+C\,a\,x-\frac {A\,a}{x}+\frac {B\,b\,x^2}{2}+\frac {C\,b\,x^3}{3}+B\,a\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 49, normalized size = 0.91 \[ - \frac {A a}{x} + B a \log {\relax (x )} + \frac {C b x^{3}}{3} + \frac {D b x^{4}}{4} + x^{2} \left (\frac {B b}{2} + \frac {D a}{2}\right ) + x \left (A b + C a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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