Integrand size = 13, antiderivative size = 21 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{5} a x^{5/2}+\frac {2}{13} c x^{13/2} \]
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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} \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{5} a x^{5/2}+\frac {2}{13} c x^{13/2} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{3/2}+c x^{11/2}\right ) \, dx \\ & = \frac {2}{5} a x^{5/2}+\frac {2}{13} c x^{13/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{65} x^{5/2} \left (13 a+5 c x^4\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {13}{2}}}{13}\) | \(14\) |
default | \(\frac {2 a \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {13}{2}}}{13}\) | \(14\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (5 x^{4} c +13 a \right )}{65}\) | \(16\) |
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{65} \, {\left (5 \, c x^{6} + 13 \, a x^{2}\right )} \sqrt {x} \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 c x^{\frac {13}{2}}}{13} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int x^{3/2} \left (a+c x^4\right ) \, dx=\frac {2\,x^{5/2}\,\left (5\,c\,x^4+13\,a\right )}{65} \]
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