Integrand size = 17, antiderivative size = 21 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{11} b x^{11/2}+\frac {2}{15} c x^{15/2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{11} b x^{11/2}+\frac {2}{15} c x^{15/2} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (b x^{9/2}+c x^{13/2}\right ) \, dx \\ & = \frac {2}{11} b x^{11/2}+\frac {2}{15} c x^{15/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{165} \left (15 b x^{11/2}+11 c x^{15/2}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {2 b \,x^{\frac {11}{2}}}{11}+\frac {2 c \,x^{\frac {15}{2}}}{15}\) | \(14\) |
default | \(\frac {2 b \,x^{\frac {11}{2}}}{11}+\frac {2 c \,x^{\frac {15}{2}}}{15}\) | \(14\) |
gosper | \(\frac {2 x^{\frac {11}{2}} \left (11 c \,x^{2}+15 b \right )}{165}\) | \(16\) |
trager | \(\frac {2 x^{\frac {11}{2}} \left (11 c \,x^{2}+15 b \right )}{165}\) | \(16\) |
risch | \(\frac {2 x^{\frac {11}{2}} \left (11 c \,x^{2}+15 b \right )}{165}\) | \(16\) |
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Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{165} \, {\left (11 \, c x^{7} + 15 \, b x^{5}\right )} \sqrt {x} \]
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Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2 b x^{\frac {11}{2}}}{11} + \frac {2 c x^{\frac {15}{2}}}{15} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{15} \, c x^{\frac {15}{2}} + \frac {2}{11} \, b x^{\frac {11}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{15} \, c x^{\frac {15}{2}} + \frac {2}{11} \, b x^{\frac {11}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int x^{5/2} \left (b x^2+c x^4\right ) \, dx=\frac {2\,x^{11/2}\,\left (11\,c\,x^2+15\,b\right )}{165} \]
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