Integrand size = 11, antiderivative size = 22 \[ \int \frac {(a+b x)^2}{x} \, dx=2 a b x+\frac {b^2 x^2}{2}+a^2 \log (x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^2}{x} \, dx=a^2 \log (x)+2 a b x+\frac {b^2 x^2}{2} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx \\ & = 2 a b x+\frac {b^2 x^2}{2}+a^2 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x} \, dx=2 a b x+\frac {b^2 x^2}{2}+a^2 \log (x) \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
default | \(2 a b x +\frac {b^{2} x^{2}}{2}+a^{2} \ln \left (x \right )\) | \(21\) |
norman | \(2 a b x +\frac {b^{2} x^{2}}{2}+a^{2} \ln \left (x \right )\) | \(21\) |
risch | \(2 a b x +\frac {b^{2} x^{2}}{2}+a^{2} \ln \left (x \right )\) | \(21\) |
parallelrisch | \(2 a b x +\frac {b^{2} x^{2}}{2}+a^{2} \ln \left (x \right )\) | \(21\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2}{x} \, dx=\frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2}{x} \, dx=a^{2} \log {\left (x \right )} + 2 a b x + \frac {b^{2} x^{2}}{2} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2}{x} \, dx=\frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^2}{x} \, dx=\frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2}{x} \, dx=a^2\,\ln \left (x\right )+\frac {b^2\,x^2}{2}+2\,a\,b\,x \]
[In]
[Out]