Integrand size = 16, antiderivative size = 55 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{5} b (A b+2 a B) x^5+\frac {1}{6} b^2 B x^6 \]
[Out]
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{5} b x^5 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{6} b^2 B x^6 \]
[In]
[Out]
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A x^2+a (2 A b+a B) x^3+b (A b+2 a B) x^4+b^2 B x^5\right ) \, dx \\ & = \frac {1}{3} a^2 A x^3+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{5} b (A b+2 a B) x^5+\frac {1}{6} b^2 B x^6 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{60} x^3 \left (5 a^2 (4 A+3 B x)+6 a b x (5 A+4 B x)+2 b^2 x^2 (6 A+5 B x)\right ) \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {b^{2} B \,x^{6}}{6}+\frac {\left (b^{2} A +2 a b B \right ) x^{5}}{5}+\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(52\) |
norman | \(\frac {b^{2} B \,x^{6}}{6}+\left (\frac {1}{5} b^{2} A +\frac {2}{5} a b B \right ) x^{5}+\left (\frac {1}{2} a b A +\frac {1}{4} a^{2} B \right ) x^{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(52\) |
gosper | \(\frac {1}{6} b^{2} B \,x^{6}+\frac {1}{5} x^{5} b^{2} A +\frac {2}{5} x^{5} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
risch | \(\frac {1}{6} b^{2} B \,x^{6}+\frac {1}{5} x^{5} b^{2} A +\frac {2}{5} x^{5} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
parallelrisch | \(\frac {1}{6} b^{2} B \,x^{6}+\frac {1}{5} x^{5} b^{2} A +\frac {2}{5} x^{5} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {A a^{2} x^{3}}{3} + \frac {B b^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2}}{5} + \frac {2 B a b}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=\frac {1}{6} \, B b^{2} x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, A a^{2} x^{3} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 (a+b x)^2 (A+B x) \, dx=x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,b^2\,x^6}{6} \]
[In]
[Out]