Optimal. Leaf size=63 \[ -\frac {a C+A b}{2 x^2}-\frac {a A}{4 x^4}-\frac {a B}{3 x^3}+\log (x) (A c+b C)-\frac {b B}{x}+B c x+\frac {1}{2} c C x^2 \]
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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} \[ -\frac {a C+A b}{2 x^2}-\frac {a A}{4 x^4}-\frac {a B}{3 x^3}+\log (x) (A c+b C)-\frac {b B}{x}+B c x+\frac {1}{2} c C x^2 \]
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^5} \, dx &=\int \left (B c+\frac {a A}{x^5}+\frac {a B}{x^4}+\frac {A b+a C}{x^3}+\frac {b B}{x^2}+\frac {A c+b C}{x}+c C x\right ) \, dx\\ &=-\frac {a A}{4 x^4}-\frac {a B}{3 x^3}-\frac {A b+a C}{2 x^2}-\frac {b B}{x}+B c x+\frac {1}{2} c C x^2+(A c+b C) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.98 \[ -\frac {a \left (3 A+4 B x+6 C x^2\right )}{12 x^4}+\frac {-A b-2 b B x+c x^3 (2 B+C x)}{2 x^2}+\log (x) (A c+b C) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 62, normalized size = 0.98 \[ \frac {6 \, C c x^{6} + 12 \, B c x^{5} + 12 \, {\left (C b + A c\right )} x^{4} \log \relax (x) - 12 \, B b x^{3} - 4 \, B a x - 6 \, {\left (C a + A b\right )} x^{2} - 3 \, A a}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 57, normalized size = 0.90 \[ \frac {1}{2} \, C c x^{2} + B c x + {\left (C b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, B b x^{3} + 4 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.92 \[ \frac {C c \,x^{2}}{2}+A c \ln \relax (x )+B c x +C b \ln \relax (x )-\frac {B b}{x}-\frac {A b}{2 x^{2}}-\frac {C a}{2 x^{2}}-\frac {B a}{3 x^{3}}-\frac {A a}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 56, normalized size = 0.89 \[ \frac {1}{2} \, C c x^{2} + B c x + {\left (C b + A c\right )} \log \relax (x) - \frac {12 \, B b x^{3} + 4 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 56, normalized size = 0.89 \[ \ln \relax (x)\,\left (A\,c+C\,b\right )-\frac {B\,b\,x^3+\left (\frac {A\,b}{2}+\frac {C\,a}{2}\right )\,x^2+\frac {B\,a\,x}{3}+\frac {A\,a}{4}}{x^4}+B\,c\,x+\frac {C\,c\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.76, size = 63, normalized size = 1.00 \[ B c x + \frac {C c x^{2}}{2} + \left (A c + C b\right ) \log {\relax (x )} + \frac {- 3 A a - 4 B a x - 12 B b x^{3} + x^{2} \left (- 6 A b - 6 C a\right )}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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