Integrand size = 28, antiderivative size = 90 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=-\frac {a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac {1}{3} b (A b+2 a C) x^3+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5+\frac {D \left (a+b x^2\right )^3}{6 b}+a^2 B \log (x) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1597, 1642} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=-\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac {D \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5 \]
[In]
[Out]
Rule 1597
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \frac {D \left (a+b x^2\right )^3}{6 b}+\int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2\right )}{x^2} \, dx \\ & = \frac {D \left (a+b x^2\right )^3}{6 b}+\int \left (a (2 A b+a C)+\frac {a^2 A}{x^2}+\frac {a^2 B}{x}+2 a b B x+b (A b+2 a C) x^2+b^2 B x^3+b^2 C x^4\right ) \, dx \\ & = -\frac {a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac {1}{3} b (A b+2 a C) x^3+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5+\frac {D \left (a+b x^2\right )^3}{6 b}+a^2 B \log (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=a^2 \left (-\frac {A}{x}+C x+\frac {D x^2}{2}\right )+\frac {1}{6} a b x (12 A+x (6 B+x (4 C+3 D x)))+\frac {1}{60} b^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+a^2 B \log (x) \]
[In]
[Out]
Time = 3.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {b^{2} D x^{6}}{6}+\frac {b^{2} C \,x^{5}}{5}+\frac {b^{2} B \,x^{4}}{4}+\frac {D a b \,x^{4}}{2}+\frac {A \,b^{2} x^{3}}{3}+\frac {2 C a b \,x^{3}}{3}+B a b \,x^{2}+\frac {D a^{2} x^{2}}{2}+2 a A b x +C \,a^{2} x +a^{2} B \ln \left (x \right )-\frac {a^{2} A}{x}\) | \(98\) |
norman | \(\frac {\left (\frac {1}{4} B \,b^{2}+\frac {1}{2} D a b \right ) x^{5}+\left (\frac {1}{3} b^{2} A +\frac {2}{3} C a b \right ) x^{4}+\left (a b B +\frac {1}{2} D a^{2}\right ) x^{3}+\left (2 a b A +C \,a^{2}\right ) x^{2}-a^{2} A +\frac {C \,b^{2} x^{6}}{5}+\frac {b^{2} D x^{7}}{6}}{x}+a^{2} B \ln \left (x \right )\) | \(100\) |
parallelrisch | \(\frac {10 b^{2} D x^{7}+12 C \,b^{2} x^{6}+15 b^{2} B \,x^{5}+30 D a b \,x^{5}+20 A \,b^{2} x^{4}+40 C a b \,x^{4}+60 B a b \,x^{3}+30 D a^{2} x^{3}+120 a A b \,x^{2}+60 a^{2} B \ln \left (x \right ) x +60 C \,a^{2} x^{2}-60 a^{2} A}{60 x}\) | \(108\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=\frac {10 \, D b^{2} x^{7} + 12 \, C b^{2} x^{6} + 15 \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + 20 \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + 60 \, B a^{2} x \log \left (x\right ) + 30 \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} - 60 \, A a^{2} + 60 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=- \frac {A a^{2}}{x} + B a^{2} \log {\left (x \right )} + \frac {C b^{2} x^{5}}{5} + \frac {D b^{2} x^{6}}{6} + x^{4} \left (\frac {B b^{2}}{4} + \frac {D a b}{2}\right ) + x^{3} \left (\frac {A b^{2}}{3} + \frac {2 C a b}{3}\right ) + x^{2} \left (B a b + \frac {D a^{2}}{2}\right ) + x \left (2 A a b + C a^{2}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=\frac {1}{6} \, D b^{2} x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {1}{4} \, {\left (2 \, D a b + B b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, C a b + A b^{2}\right )} x^{3} + B a^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (D a^{2} + 2 \, B a b\right )} x^{2} - \frac {A a^{2}}{x} + {\left (C a^{2} + 2 \, A a b\right )} x \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=\frac {1}{6} \, D b^{2} x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {1}{2} \, D a b x^{4} + \frac {1}{4} \, B b^{2} x^{4} + \frac {2}{3} \, C a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{2} \, D a^{2} x^{2} + B a b x^{2} + C a^{2} x + 2 \, A a b x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac {A a^{2}}{x} \]
[In]
[Out]
Time = 5.56 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx=\frac {B\,\left (4\,a^2\,\ln \left (x\right )+b^2\,x^4+4\,a\,b\,x^2\right )}{4}+\frac {{\left (b\,x^2+a\right )}^3\,D}{6\,b}+\frac {C\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {A\,\left (-3\,a^2+6\,a\,b\,x^2+b^2\,x^4\right )}{3\,x} \]
[In]
[Out]