Optimal. Leaf size=63 \[ \log (x) (a C+A b)-\frac {a A}{2 x^2}-\frac {a B}{x}+\frac {1}{2} x^2 (A c+b C)+b B x+\frac {1}{3} B c x^3+\frac {1}{4} c C x^4 \]
[Out]
________________________________________________________________________________________
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} \[ \log (x) (a C+A b)-\frac {a A}{2 x^2}-\frac {a B}{x}+\frac {1}{2} x^2 (A c+b C)+b B x+\frac {1}{3} B c x^3+\frac {1}{4} c C x^4 \]
Antiderivative was successfully verified.
[In]
[Out]
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^3} \, dx &=\int \left (b B+\frac {a A}{x^3}+\frac {a B}{x^2}+\frac {A b+a C}{x}+(A c+b C) x+B c x^2+c C x^3\right ) \, dx\\ &=-\frac {a A}{2 x^2}-\frac {a B}{x}+b B x+\frac {1}{2} (A c+b C) x^2+\frac {1}{3} B c x^3+\frac {1}{4} c C x^4+(A b+a C) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 58, normalized size = 0.92 \[ \log (x) (a C+A b)-\frac {a (A+2 B x)}{2 x^2}+\frac {1}{12} x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 62, normalized size = 0.98 \[ \frac {3 \, C c x^{6} + 4 \, B c x^{5} + 12 \, B b x^{3} + 6 \, {\left (C b + A c\right )} x^{4} + 12 \, {\left (C a + A b\right )} x^{2} \log \relax (x) - 12 \, B a x - 6 \, A a}{12 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 58, normalized size = 0.92 \[ \frac {1}{4} \, C c x^{4} + \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, C b x^{2} + \frac {1}{2} \, A c x^{2} + B b x + {\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 58, normalized size = 0.92 \[ \frac {C c \,x^{4}}{4}+\frac {B c \,x^{3}}{3}+\frac {A c \,x^{2}}{2}+\frac {C b \,x^{2}}{2}+A b \ln \relax (x )+B b x +C a \ln \relax (x )-\frac {B a}{x}-\frac {A a}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.75, size = 55, normalized size = 0.87 \[ \frac {1}{4} \, C c x^{4} + \frac {1}{3} \, B c x^{3} + B b x + \frac {1}{2} \, {\left (C b + A c\right )} x^{2} + {\left (C a + A b\right )} \log \relax (x) - \frac {2 \, B a x + A a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 56, normalized size = 0.89 \[ x^2\,\left (\frac {A\,c}{2}+\frac {C\,b}{2}\right )-\frac {\frac {A\,a}{2}+B\,a\,x}{x^2}+\ln \relax (x)\,\left (A\,b+C\,a\right )+B\,b\,x+\frac {B\,c\,x^3}{3}+\frac {C\,c\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.29, size = 61, normalized size = 0.97 \[ B b x + \frac {B c x^{3}}{3} + \frac {C c x^{4}}{4} + x^{2} \left (\frac {A c}{2} + \frac {C b}{2}\right ) + \left (A b + C a\right ) \log {\relax (x )} + \frac {- A a - 2 B a x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________