Integrand size = 15, antiderivative size = 32 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {2 b^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 a^2 \sqrt {x} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 a^2 \sqrt {x}-\frac {4 a b}{\sqrt {x}}-\frac {2 b^2}{3 x^{3/2}} \]
[In]
[Out]
Rule 45
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^2}{x^{5/2}} \, dx \\ & = \int \left (\frac {b^2}{x^{5/2}}+\frac {2 a b}{x^{3/2}}+\frac {a^2}{\sqrt {x}}\right ) \, dx \\ & = -\frac {2 b^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 a^2 \sqrt {x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {2 \left (b^2+6 a b x-3 a^2 x^2\right )}{3 x^{3/2}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\) | \(25\) |
derivativedivides | \(-\frac {2 b^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 a^{2} \sqrt {x}\) | \(25\) |
default | \(-\frac {2 b^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 a^{2} \sqrt {x}\) | \(25\) |
trager | \(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\) | \(25\) |
risch | \(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\) | \(25\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=\frac {2 \, {\left (3 \, a^{2} x^{2} - 6 \, a b x - b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 a^{2} \sqrt {x} - \frac {4 a b}{\sqrt {x}} - \frac {2 b^{2}}{3 x^{\frac {3}{2}}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 \, a^{2} \sqrt {x} - \frac {4 \, a b}{\sqrt {x}} - \frac {2 \, b^{2}}{3 \, x^{\frac {3}{2}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 \, a^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
[In]
[Out]
Time = 5.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {-6\,a^2\,x^2+12\,a\,b\,x+2\,b^2}{3\,x^{3/2}} \]
[In]
[Out]