Integrand size = 15, antiderivative size = 29 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {(a+b x)^2}{2 a c x \sqrt {c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 37} \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {(a+b x)^2}{2 a c x \sqrt {c x^2}} \]
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Rule 15
Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {a+b x}{x^3} \, dx}{c \sqrt {c x^2}} \\ & = -\frac {(a+b x)^2}{2 a c x \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {x (a+2 b x)}{2 \left (c x^2\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {x \left (2 b x +a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(17\) |
default | \(-\frac {x \left (2 b x +a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(17\) |
risch | \(\frac {-b x -\frac {a}{2}}{c x \sqrt {c \,x^{2}}}\) | \(23\) |
trager | \(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 c^{2} x^{3}}\) | \(28\) |
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Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=- \frac {a x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {b}{\sqrt {c x^{2}} c} - \frac {a}{2 \, c^{\frac {3}{2}} x^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {2 \, b x + a}{2 \, c^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right )} \]
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Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x}{\left (c x^2\right )^{3/2}} \, dx=-\frac {2\,b\,x^3+a\,x^2}{2\,c^{3/2}\,x\,{\left (x^2\right )}^{3/2}} \]
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