Integrand size = 19, antiderivative size = 36 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1598, 276} \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \]
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Rule 276
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {x} \left (b+c x^2\right )^2 \, dx \\ & = \int \left (b^2 \sqrt {x}+2 b c x^{5/2}+c^2 x^{9/2}\right ) \, dx \\ & = \frac {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{231} x^{3/2} \left (77 b^2+66 b c x^2+21 c^2 x^4\right ) \]
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Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3}+\frac {4 b c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {11}{2}}}{11}\) | \(25\) |
default | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3}+\frac {4 b c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {11}{2}}}{11}\) | \(25\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (21 c^{2} x^{4}+66 b c \,x^{2}+77 b^{2}\right )}{231}\) | \(27\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (21 c^{2} x^{4}+66 b c \,x^{2}+77 b^{2}\right )}{231}\) | \(27\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (21 c^{2} x^{4}+66 b c \,x^{2}+77 b^{2}\right )}{231}\) | \(27\) |
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{231} \, {\left (21 \, c^{2} x^{5} + 66 \, b c x^{3} + 77 \, b^{2} x\right )} \sqrt {x} \]
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Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2 b^{2} x^{\frac {3}{2}}}{3} + \frac {4 b c x^{\frac {7}{2}}}{7} + \frac {2 c^{2} x^{\frac {11}{2}}}{11} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} x^{\frac {3}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} x^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx=\frac {2\,x^{3/2}\,\left (77\,b^2+66\,b\,c\,x^2+21\,c^2\,x^4\right )}{231} \]
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