Integrand size = 18, antiderivative size = 35 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {a+b x}{x^4} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^4}+\frac {b}{x^3}\right ) \, dx}{\sqrt {c x^2}} \\ & = -\frac {a}{3 x^2 \sqrt {c x^2}}-\frac {b}{2 x \sqrt {c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {c (2 a+3 b x)}{6 \left (c x^2\right )^{3/2}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {-\frac {b x}{2}-\frac {a}{3}}{x^{2} \sqrt {c \,x^{2}}}\) | \(20\) |
gosper | \(-\frac {3 b x +2 a}{6 x^{2} \sqrt {c \,x^{2}}}\) | \(21\) |
default | \(-\frac {3 b x +2 a}{6 x^{2} \sqrt {c \,x^{2}}}\) | \(21\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a \,x^{2}+3 b \,x^{2}+2 a x +3 b x +2 a \right ) \sqrt {c \,x^{2}}}{6 c \,x^{4}}\) | \(43\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (3 \, b x + 2 \, a\right )}}{6 \, c x^{4}} \]
[In]
[Out]
Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=- \frac {a}{3 x^{2} \sqrt {c x^{2}}} - \frac {b}{2 x \sqrt {c x^{2}}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {b}{2 \, \sqrt {c} x^{2}} - \frac {a}{3 \, \sqrt {c} x^{3}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {3 \, b x + 2 \, a}{6 \, \sqrt {c} x^{3} \mathrm {sgn}\left (x\right )} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x}{x^3 \sqrt {c x^2}} \, dx=-\frac {2\,a\,\sqrt {x^2}+3\,b\,x\,\sqrt {x^2}}{6\,\sqrt {c}\,x^4} \]
[In]
[Out]