Integrand size = 13, antiderivative size = 21 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=-\frac {2 a}{5 x^{5/2}}+\frac {2}{3} c x^{3/2} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=\frac {2}{3} c x^{3/2}-\frac {2 a}{5 x^{5/2}} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x^{7/2}}+c \sqrt {x}\right ) \, dx \\ & = -\frac {2 a}{5 x^{5/2}}+\frac {2}{3} c x^{3/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=-\frac {2 \left (3 a-5 c x^4\right )}{15 x^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(-\frac {2 a}{5 x^{\frac {5}{2}}}+\frac {2 c \,x^{\frac {3}{2}}}{3}\) | \(14\) |
default | \(-\frac {2 a}{5 x^{\frac {5}{2}}}+\frac {2 c \,x^{\frac {3}{2}}}{3}\) | \(14\) |
gosper | \(-\frac {2 \left (-5 x^{4} c +3 a \right )}{15 x^{\frac {5}{2}}}\) | \(16\) |
trager | \(-\frac {2 \left (-5 x^{4} c +3 a \right )}{15 x^{\frac {5}{2}}}\) | \(16\) |
risch | \(-\frac {2 \left (-5 x^{4} c +3 a \right )}{15 x^{\frac {5}{2}}}\) | \(16\) |
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=\frac {2 \, {\left (5 \, c x^{4} - 3 \, a\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=- \frac {2 a}{5 x^{\frac {5}{2}}} + \frac {2 c x^{\frac {3}{2}}}{3} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=\frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, a}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=\frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, a}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {a+c x^4}{x^{7/2}} \, dx=-\frac {6\,a-10\,c\,x^4}{15\,x^{5/2}} \]
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