Integrand size = 16, antiderivative size = 22 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = a^2 x+a b x^2+\frac {b^2 x^3}{3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\left (b x +a \right )^{3}}{3 b}\) | \(13\) |
norman | \(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\) | \(21\) |
risch | \(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\) | \(21\) |
parallelrisch | \(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\) | \(21\) |
parts | \(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\) | \(21\) |
gosper | \(\frac {x \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3}\) | \(22\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2\,x+a\,b\,x^2+\frac {b^2\,x^3}{3} \]
[In]
[Out]