Integrand size = 15, antiderivative size = 41 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a}{4 c^2 x^3 \sqrt {c x^2}}-\frac {b}{3 c^2 x^2 \sqrt {c x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 41, 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 = {15, 45} \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a}{4 c^2 x^3 \sqrt {c x^2}}-\frac {b}{3 c^2 x^2 \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {a+b x}{x^5} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx}{c^2 \sqrt {c x^2}} \\ & = -\frac {a}{4 c^2 x^3 \sqrt {c x^2}}-\frac {b}{3 c^2 x^2 \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {x (3 a+4 b x)}{12 \left (c x^2\right )^{5/2}} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {x \left (4 b x +3 a \right )}{12 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(19\) |
default | \(-\frac {x \left (4 b x +3 a \right )}{12 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(19\) |
risch | \(\frac {-\frac {b x}{3}-\frac {a}{4}}{c^{2} x^{3} \sqrt {c \,x^{2}}}\) | \(23\) |
trager | \(\frac {\left (-1+x \right ) \left (3 a \,x^{3}+4 b \,x^{3}+3 a \,x^{2}+4 b \,x^{2}+3 a x +4 b x +3 a \right ) \sqrt {c \,x^{2}}}{12 c^{3} x^{5}}\) | \(55\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (4 \, b x + 3 \, a\right )}}{12 \, c^{3} x^{5}} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=- \frac {a x}{4 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b x^{2}}{3 \left (c x^{2}\right )^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {b}{3 \, \left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a}{4 \, c^{\frac {5}{2}} x^{4}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.49 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {4 \, b x + 3 \, a}{12 \, c^{\frac {5}{2}} x^{4} \mathrm {sgn}\left (x\right )} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{\left (c x^2\right )^{5/2}} \, dx=-\frac {3\,a\,\sqrt {x^2}+4\,b\,x\,\sqrt {x^2}}{12\,c^{5/2}\,x^5} \]
[In]
[Out]