Integrand size = 26, antiderivative size = 63 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=-\frac {a A}{4 x^4}-\frac {a B}{3 x^3}-\frac {A b+a C}{2 x^2}-\frac {b B}{x}+B c x+\frac {1}{2} c C x^2+(A c+b C) \log (x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1642} \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=-\frac {a C+A b}{2 x^2}-\frac {a A}{4 x^4}-\frac {a B}{3 x^3}+\log (x) (A c+b C)-\frac {b B}{x}+B c x+\frac {1}{2} c C x^2 \]
[In]
[Out]
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=-\frac {a \left (3 A+4 B x+6 C x^2\right )}{12 x^4}+\frac {-A b-2 b B x+c x^3 (2 B+C x)}{2 x^2}+(A c+b C) \log (x) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {c C \,x^{2}}{2}+B c x +\left (A c +C b \right ) \ln \left (x \right )-\frac {A b +C a}{2 x^{2}}-\frac {b B}{x}-\frac {a A}{4 x^{4}}-\frac {a B}{3 x^{3}}\) | \(56\) |
risch | \(\frac {c C \,x^{2}}{2}+B c x +\frac {-B b \,x^{3}+\left (-\frac {A b}{2}-\frac {C a}{2}\right ) x^{2}-\frac {B a x}{3}-\frac {A a}{4}}{x^{4}}+A \ln \left (x \right ) c +C \ln \left (x \right ) b\) | \(57\) |
norman | \(\frac {\left (-\frac {A b}{2}-\frac {C a}{2}\right ) x^{2}+B c \,x^{5}-\frac {A a}{4}-\frac {B a x}{3}-B b \,x^{3}+\frac {c C \,x^{6}}{2}}{x^{4}}+\left (A c +C b \right ) \ln \left (x \right )\) | \(59\) |
parallelrisch | \(\frac {6 c C \,x^{6}+12 A \ln \left (x \right ) x^{4} c +12 B c \,x^{5}+12 C \ln \left (x \right ) x^{4} b -12 B b \,x^{3}-6 A b \,x^{2}-6 C a \,x^{2}-4 B a x -3 A a}{12 x^{4}}\) | \(69\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=\frac {6 \, C c x^{6} + 12 \, B c x^{5} + 12 \, {\left (C b + A c\right )} x^{4} \log \left (x\right ) - 12 \, B b x^{3} - 4 \, B a x - 6 \, {\left (C a + A b\right )} x^{2} - 3 \, A a}{12 \, x^{4}} \]
[In]
[Out]
Time = 0.96 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=B c x + \frac {C c x^{2}}{2} + \left (A c + C b\right ) \log {\left (x \right )} + \frac {- 3 A a - 4 B a x - 12 B b x^{3} + x^{2} \left (- 6 A b - 6 C a\right )}{12 x^{4}} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=\frac {1}{2} \, C c x^{2} + B c x + {\left (C b + A c\right )} \log \left (x\right ) - \frac {12 \, B b x^{3} + 4 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=\frac {1}{2} \, C c x^{2} + B c x + {\left (C b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, B b x^{3} + 4 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^5} \, dx=\ln \left (x\right )\,\left (A\,c+C\,b\right )-\frac {B\,b\,x^3+\left (\frac {A\,b}{2}+\frac {C\,a}{2}\right )\,x^2+\frac {B\,a\,x}{3}+\frac {A\,a}{4}}{x^4}+B\,c\,x+\frac {C\,c\,x^2}{2} \]
[In]
[Out]