Integrand size = 19, antiderivative size = 51 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{13} b^3 x^{13/2}+\frac {6}{17} b^2 c x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \]
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Time = 0.01 (sec) , antiderivative size = 51, 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} \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{13} b^3 x^{13/2}+\frac {6}{17} b^2 c x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \]
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Rule 276
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x^{11/2} \left (b+c x^2\right )^3 \, dx \\ & = \int \left (b^3 x^{11/2}+3 b^2 c x^{15/2}+3 b c^2 x^{19/2}+c^3 x^{23/2}\right ) \, dx \\ & = \frac {2}{13} b^3 x^{13/2}+\frac {6}{17} b^2 c x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2 x^{13/2} \left (2975 b^3+6825 b^2 c x^2+5525 b c^2 x^4+1547 c^3 x^6\right )}{38675} \]
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Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 b^{3} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,x^{\frac {17}{2}}}{17}+\frac {2 b \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}\) | \(36\) |
default | \(\frac {2 b^{3} x^{\frac {13}{2}}}{13}+\frac {6 b^{2} c \,x^{\frac {17}{2}}}{17}+\frac {2 b \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 c^{3} x^{\frac {25}{2}}}{25}\) | \(36\) |
gosper | \(\frac {2 x^{\frac {13}{2}} \left (1547 c^{3} x^{6}+5525 b \,c^{2} x^{4}+6825 b^{2} c \,x^{2}+2975 b^{3}\right )}{38675}\) | \(38\) |
trager | \(\frac {2 x^{\frac {13}{2}} \left (1547 c^{3} x^{6}+5525 b \,c^{2} x^{4}+6825 b^{2} c \,x^{2}+2975 b^{3}\right )}{38675}\) | \(38\) |
risch | \(\frac {2 x^{\frac {13}{2}} \left (1547 c^{3} x^{6}+5525 b \,c^{2} x^{4}+6825 b^{2} c \,x^{2}+2975 b^{3}\right )}{38675}\) | \(38\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{38675} \, {\left (1547 \, c^{3} x^{12} + 5525 \, b c^{2} x^{10} + 6825 \, b^{2} c x^{8} + 2975 \, b^{3} x^{6}\right )} \sqrt {x} \]
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Time = 0.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2 b^{3} x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \frac {\left (b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2\,b^3\,x^{13/2}}{13}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {6\,b^2\,c\,x^{17/2}}{17}+\frac {2\,b\,c^2\,x^{21/2}}{7} \]
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