Integrand size = 18, antiderivative size = 41 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \sqrt {c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {a+b x}{x^6} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^6}+\frac {b}{x^5}\right ) \, dx}{c^2 \sqrt {c x^2}} \\ & = -\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \sqrt {c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {c x^2 (4 a+5 b x)}{20 \left (c x^2\right )^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {5 b x +4 a}{20 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(18\) |
default | \(-\frac {5 b x +4 a}{20 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(18\) |
risch | \(\frac {-\frac {b x}{4}-\frac {a}{5}}{c^{2} x^{4} \sqrt {c \,x^{2}}}\) | \(23\) |
trager | \(\frac {\left (-1+x \right ) \left (4 a \,x^{4}+5 b \,x^{4}+4 a \,x^{3}+5 b \,x^{3}+4 a \,x^{2}+5 b \,x^{2}+4 a x +5 b x +4 a \right ) \sqrt {c \,x^{2}}}{20 c^{3} x^{6}}\) | \(67\) |
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Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (5 \, b x + 4 \, a\right )}}{20 \, c^{3} x^{6}} \]
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Time = 0.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=- \frac {a}{5 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b x}{4 \left (c x^{2}\right )^{\frac {5}{2}}} \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {b}{4 \, c^{\frac {5}{2}} x^{4}} - \frac {a}{5 \, c^{\frac {5}{2}} x^{5}} \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.49 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {5 \, b x + 4 \, a}{20 \, c^{\frac {5}{2}} x^{5} \mathrm {sgn}\left (x\right )} \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx=-\frac {4\,a\,\sqrt {x^2}+5\,b\,x\,\sqrt {x^2}}{20\,c^{5/2}\,x^6} \]
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