Integrand size = 15, antiderivative size = 49 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \]
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Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3}{\sqrt {x}}+3 a^2 c x^{7/2}+3 a c^2 x^{15/2}+c^3 x^{23/2}\right ) \, dx \\ & = 2 a^3 \sqrt {x}+\frac {2}{3} a^2 c x^{9/2}+\frac {6}{17} a c^2 x^{17/2}+\frac {2}{25} c^3 x^{25/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (1275 a^3+425 a^2 c x^4+225 a c^2 x^8+51 c^3 x^{12}\right )}{1275} \]
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Time = 3.94 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {2 a^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}+2 a^{3} \sqrt {x}\) | \(36\) |
default | \(\frac {2 a^{2} c \,x^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}+2 a^{3} \sqrt {x}\) | \(36\) |
trager | \(\left (\frac {2}{25} c^{3} x^{12}+\frac {6}{17} a \,c^{2} x^{8}+\frac {2}{3} a^{2} c \,x^{4}+2 a^{3}\right ) \sqrt {x}\) | \(37\) |
gosper | \(\frac {2 \sqrt {x}\, \left (51 c^{3} x^{12}+225 a \,c^{2} x^{8}+425 a^{2} c \,x^{4}+1275 a^{3}\right )}{1275}\) | \(38\) |
risch | \(\frac {2 \sqrt {x}\, \left (51 c^{3} x^{12}+225 a \,c^{2} x^{8}+425 a^{2} c \,x^{4}+1275 a^{3}\right )}{1275}\) | \(38\) |
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{1275} \, {\left (51 \, c^{3} x^{12} + 225 \, a c^{2} x^{8} + 425 \, a^{2} c x^{4} + 1275 \, a^{3}\right )} \sqrt {x} \]
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Time = 0.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^{3} \sqrt {x} + \frac {2 a^{2} c x^{\frac {9}{2}}}{3} + \frac {6 a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + 2 \, a^{3} \sqrt {x} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+c x^4\right )^3}{\sqrt {x}} \, dx=2\,a^3\,\sqrt {x}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {2\,a^2\,c\,x^{9/2}}{3}+\frac {6\,a\,c^2\,x^{17/2}}{17} \]
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