Optimal. Leaf size=63 \[ x (a C+A b)-\frac {a A}{x}+a B \log (x)+\frac {1}{3} x^3 (A c+b C)+\frac {1}{2} b B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \]
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Rubi [A] time = 0.05, antiderivative size = 63, 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 = {1628} \begin {gather*} x (a C+A b)-\frac {a A}{x}+a B \log (x)+\frac {1}{3} x^3 (A c+b C)+\frac {1}{2} b B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\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 x+(A c+b C) x^2+B c x^3+c C x^4\right ) \, dx\\ &=-\frac {a A}{x}+(A b+a C) x+\frac {1}{2} b B x^2+\frac {1}{3} (A c+b C) x^3+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5+a B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 1.00 \begin {gather*} x (a C+A b)-\frac {a A}{x}+a B \log (x)+\frac {1}{3} x^3 (A c+b C)+\frac {1}{2} b B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.23, size = 62, normalized size = 0.98 \begin {gather*} \frac {12 \, C c x^{6} + 15 \, B c x^{5} + 30 \, B b x^{3} + 20 \, {\left (C b + A c\right )} x^{4} + 60 \, B a x \log \relax (x) + 60 \, {\left (C a + A b\right )} x^{2} - 60 \, A a}{60 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 57, normalized size = 0.90 \begin {gather*} \frac {1}{5} \, C c x^{5} + \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{3} \, A c x^{3} + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.90 \begin {gather*} \frac {C c \,x^{5}}{5}+\frac {B c \,x^{4}}{4}+\frac {A c \,x^{3}}{3}+\frac {C b \,x^{3}}{3}+\frac {B b \,x^{2}}{2}+A b x +B a \ln \relax (x )+C a x -\frac {A a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 55, normalized size = 0.87 \begin {gather*} \frac {1}{5} \, C c x^{5} + \frac {1}{4} \, B c x^{4} + \frac {1}{2} \, B b x^{2} + \frac {1}{3} \, {\left (C b + A c\right )} x^{3} + B a \log \relax (x) + {\left (C a + A b\right )} x - \frac {A a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 56, normalized size = 0.89 \begin {gather*} x\,\left (A\,b+C\,a\right )+x^3\,\left (\frac {A\,c}{3}+\frac {C\,b}{3}\right )-\frac {A\,a}{x}+\frac {B\,b\,x^2}{2}+\frac {B\,c\,x^4}{4}+\frac {C\,c\,x^5}{5}+B\,a\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 58, normalized size = 0.92 \begin {gather*} - \frac {A a}{x} + B a \log {\relax (x )} + \frac {B b x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {C c x^{5}}{5} + x^{3} \left (\frac {A c}{3} + \frac {C b}{3}\right ) + x \left (A b + C a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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