Integrand size = 15, antiderivative size = 36 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {276} \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x^{5/2}+2 a c x^{13/2}+c^2 x^{21/2}\right ) \, dx \\ & = \frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2 x^{7/2} \left (345 a^2+322 a c x^4+105 c^2 x^8\right )}{2415} \]
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Time = 3.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {2 a^{2} x^{\frac {7}{2}}}{7}+\frac {4 a c \,x^{\frac {15}{2}}}{15}+\frac {2 c^{2} x^{\frac {23}{2}}}{23}\) | \(25\) |
default | \(\frac {2 a^{2} x^{\frac {7}{2}}}{7}+\frac {4 a c \,x^{\frac {15}{2}}}{15}+\frac {2 c^{2} x^{\frac {23}{2}}}{23}\) | \(25\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (105 c^{2} x^{8}+322 a \,x^{4} c +345 a^{2}\right )}{2415}\) | \(27\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (105 c^{2} x^{8}+322 a \,x^{4} c +345 a^{2}\right )}{2415}\) | \(27\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (105 c^{2} x^{8}+322 a \,x^{4} c +345 a^{2}\right )}{2415}\) | \(27\) |
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none
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2}{2415} \, {\left (105 \, c^{2} x^{11} + 322 \, a c x^{7} + 345 \, a^{2} x^{3}\right )} \sqrt {x} \]
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Time = 0.88 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2 a^{2} x^{\frac {7}{2}}}{7} + \frac {4 a c x^{\frac {15}{2}}}{15} + \frac {2 c^{2} x^{\frac {23}{2}}}{23} \]
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none
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{15} \, a c x^{\frac {15}{2}} + \frac {2}{7} \, a^{2} x^{\frac {7}{2}} \]
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none
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=\frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{15} \, a c x^{\frac {15}{2}} + \frac {2}{7} \, a^{2} x^{\frac {7}{2}} \]
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Time = 5.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int x^{5/2} \left (a+c x^4\right )^2 \, dx=x^{7/2}\,\left (\frac {2\,a^2}{7}+\frac {4\,a\,c\,x^4}{15}+\frac {2\,c^2\,x^8}{23}\right ) \]
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