Integrand size = 19, antiderivative size = 51 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2}{15} b^3 x^{15/2}+\frac {6}{19} b^2 c x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]
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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} \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2}{15} b^3 x^{15/2}+\frac {6}{19} b^2 c x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]
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Rule 276
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x^{13/2} \left (b+c x^2\right )^3 \, dx \\ & = \int \left (b^3 x^{13/2}+3 b^2 c x^{17/2}+3 b c^2 x^{21/2}+c^3 x^{25/2}\right ) \, dx \\ & = \frac {2}{15} b^3 x^{15/2}+\frac {6}{19} b^2 c x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2 x^{15/2} \left (3933 b^3+9315 b^2 c x^2+7695 b c^2 x^4+2185 c^3 x^6\right )}{58995} \]
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Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 b^{3} x^{\frac {15}{2}}}{15}+\frac {6 b^{2} c \,x^{\frac {19}{2}}}{19}+\frac {6 b \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 c^{3} x^{\frac {27}{2}}}{27}\) | \(36\) |
default | \(\frac {2 b^{3} x^{\frac {15}{2}}}{15}+\frac {6 b^{2} c \,x^{\frac {19}{2}}}{19}+\frac {6 b \,c^{2} x^{\frac {23}{2}}}{23}+\frac {2 c^{3} x^{\frac {27}{2}}}{27}\) | \(36\) |
gosper | \(\frac {2 x^{\frac {15}{2}} \left (2185 c^{3} x^{6}+7695 b \,c^{2} x^{4}+9315 b^{2} c \,x^{2}+3933 b^{3}\right )}{58995}\) | \(38\) |
trager | \(\frac {2 x^{\frac {15}{2}} \left (2185 c^{3} x^{6}+7695 b \,c^{2} x^{4}+9315 b^{2} c \,x^{2}+3933 b^{3}\right )}{58995}\) | \(38\) |
risch | \(\frac {2 x^{\frac {15}{2}} \left (2185 c^{3} x^{6}+7695 b \,c^{2} x^{4}+9315 b^{2} c \,x^{2}+3933 b^{3}\right )}{58995}\) | \(38\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2}{58995} \, {\left (2185 \, c^{3} x^{13} + 7695 \, b c^{2} x^{11} + 9315 \, b^{2} c x^{9} + 3933 \, b^{3} x^{7}\right )} \sqrt {x} \]
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Time = 1.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2 b^{3} x^{\frac {15}{2}}}{15} + \frac {6 b^{2} c x^{\frac {19}{2}}}{19} + \frac {6 b c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx=\frac {2\,b^3\,x^{15/2}}{15}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,b^2\,c\,x^{19/2}}{19}+\frac {6\,b\,c^2\,x^{23/2}}{23} \]
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