Integrand size = 15, antiderivative size = 34 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=-\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2} \]
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Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=-\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{x^{3/2}}+2 a c x^{5/2}+c^2 x^{13/2}\right ) \, dx \\ & = -\frac {2 a^2}{\sqrt {x}}+\frac {4}{7} a c x^{7/2}+\frac {2}{15} c^2 x^{15/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=-\frac {2 \left (105 a^2-30 a c x^4-7 c^2 x^8\right )}{105 \sqrt {x}} \]
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Time = 3.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {15}{2}}}{15}-\frac {2 a^{2}}{\sqrt {x}}\) | \(25\) |
default | \(\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 c^{2} x^{\frac {15}{2}}}{15}-\frac {2 a^{2}}{\sqrt {x}}\) | \(25\) |
gosper | \(-\frac {2 \left (-7 c^{2} x^{8}-30 a \,x^{4} c +105 a^{2}\right )}{105 \sqrt {x}}\) | \(27\) |
trager | \(-\frac {2 \left (-7 c^{2} x^{8}-30 a \,x^{4} c +105 a^{2}\right )}{105 \sqrt {x}}\) | \(27\) |
risch | \(-\frac {2 \left (-7 c^{2} x^{8}-30 a \,x^{4} c +105 a^{2}\right )}{105 \sqrt {x}}\) | \(27\) |
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none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=\frac {2 \, {\left (7 \, c^{2} x^{8} + 30 \, a c x^{4} - 105 \, a^{2}\right )}}{105 \, \sqrt {x}} \]
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Time = 0.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=- \frac {2 a^{2}}{\sqrt {x}} + \frac {4 a c x^{\frac {7}{2}}}{7} + \frac {2 c^{2} x^{\frac {15}{2}}}{15} \]
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none
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=\frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{7} \, a c x^{\frac {7}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]
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none
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=\frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{7} \, a c x^{\frac {7}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+c x^4\right )^2}{x^{3/2}} \, dx=\frac {-2\,a^2+\frac {4\,a\,c\,x^4}{7}+\frac {2\,c^2\,x^8}{15}}{\sqrt {x}} \]
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