Integrand size = 20, antiderivative size = 103 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a \left (b^2+a c\right ) x^{15/2}+\frac {2}{19} b \left (b^2+6 a c\right ) x^{19/2}+\frac {6}{23} c \left (b^2+a c\right ) x^{23/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \]
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Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122} \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {6}{23} c x^{23/2} \left (a c+b^2\right )+\frac {2}{19} b x^{19/2} \left (6 a c+b^2\right )+\frac {2}{5} a x^{15/2} \left (a c+b^2\right )+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \]
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Rule 1122
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^{5/2}+3 a^2 b x^{9/2}+3 a \left (b^2+a c\right ) x^{13/2}+b \left (b^2+6 a c\right ) x^{17/2}+3 c \left (b^2+a c\right ) x^{21/2}+3 b c^2 x^{25/2}+c^3 x^{29/2}\right ) \, dx \\ & = \frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a \left (b^2+a c\right ) x^{15/2}+\frac {2}{19} b \left (b^2+6 a c\right ) x^{19/2}+\frac {6}{23} c \left (b^2+a c\right ) x^{23/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 \left (6705765 a^3 x^{7/2}+12801915 a^2 b x^{11/2}+9388071 a b^2 x^{15/2}+9388071 a^2 c x^{15/2}+2470545 b^3 x^{19/2}+14823270 a b c x^{19/2}+6122655 b^2 c x^{23/2}+6122655 a c^2 x^{23/2}+5215595 b c^2 x^{27/2}+1514205 c^3 x^{31/2}\right )}{46940355} \]
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Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (1514205 c^{3} x^{12}+5215595 b \,c^{2} x^{10}+6122655 a \,c^{2} x^{8}+6122655 b^{2} c \,x^{8}+14823270 a b c \,x^{6}+2470545 b^{3} x^{6}+9388071 a^{2} c \,x^{4}+9388071 b^{2} x^{4} a +12801915 a^{2} b \,x^{2}+6705765 a^{3}\right )}{46940355}\) | \(90\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (1514205 c^{3} x^{12}+5215595 b \,c^{2} x^{10}+6122655 a \,c^{2} x^{8}+6122655 b^{2} c \,x^{8}+14823270 a b c \,x^{6}+2470545 b^{3} x^{6}+9388071 a^{2} c \,x^{4}+9388071 b^{2} x^{4} a +12801915 a^{2} b \,x^{2}+6705765 a^{3}\right )}{46940355}\) | \(90\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (1514205 c^{3} x^{12}+5215595 b \,c^{2} x^{10}+6122655 a \,c^{2} x^{8}+6122655 b^{2} c \,x^{8}+14823270 a b c \,x^{6}+2470545 b^{3} x^{6}+9388071 a^{2} c \,x^{4}+9388071 b^{2} x^{4} a +12801915 a^{2} b \,x^{2}+6705765 a^{3}\right )}{46940355}\) | \(90\) |
derivativedivides | \(\frac {2 c^{3} x^{\frac {31}{2}}}{31}+\frac {2 b \,c^{2} x^{\frac {27}{2}}}{9}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {6 a^{2} b \,x^{\frac {11}{2}}}{11}+\frac {2 a^{3} x^{\frac {7}{2}}}{7}\) | \(111\) |
default | \(\frac {2 c^{3} x^{\frac {31}{2}}}{31}+\frac {2 b \,c^{2} x^{\frac {27}{2}}}{9}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {6 a^{2} b \,x^{\frac {11}{2}}}{11}+\frac {2 a^{3} x^{\frac {7}{2}}}{7}\) | \(111\) |
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{46940355} \, {\left (1514205 \, c^{3} x^{15} + 5215595 \, b c^{2} x^{13} + 6122655 \, {\left (b^{2} c + a c^{2}\right )} x^{11} + 2470545 \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + 12801915 \, a^{2} b x^{5} + 9388071 \, {\left (a b^{2} + a^{2} c\right )} x^{7} + 6705765 \, a^{3} x^{3}\right )} \sqrt {x} \]
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Time = 1.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {6 a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 a^{2} c x^{\frac {15}{2}}}{5} + \frac {2 a b^{2} x^{\frac {15}{2}}}{5} + \frac {12 a b c x^{\frac {19}{2}}}{19} + \frac {6 a c^{2} x^{\frac {23}{2}}}{23} + \frac {2 b^{3} x^{\frac {19}{2}}}{19} + \frac {6 b^{2} c x^{\frac {23}{2}}}{23} + \frac {2 b c^{2} x^{\frac {27}{2}}}{9} + \frac {2 c^{3} x^{\frac {31}{2}}}{31} \]
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Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {2}{9} \, b c^{2} x^{\frac {27}{2}} + \frac {6}{23} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {19}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{5} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {15}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=\frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {2}{9} \, b c^{2} x^{\frac {27}{2}} + \frac {6}{23} \, b^{2} c x^{\frac {23}{2}} + \frac {6}{23} \, a c^{2} x^{\frac {23}{2}} + \frac {2}{19} \, b^{3} x^{\frac {19}{2}} + \frac {12}{19} \, a b c x^{\frac {19}{2}} + \frac {2}{5} \, a b^{2} x^{\frac {15}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {15}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx=x^{19/2}\,\left (\frac {2\,b^3}{19}+\frac {12\,a\,c\,b}{19}\right )+\frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,c^3\,x^{31/2}}{31}+\frac {6\,a^2\,b\,x^{11/2}}{11}+\frac {2\,b\,c^2\,x^{27/2}}{9}+\frac {2\,a\,x^{15/2}\,\left (b^2+a\,c\right )}{5}+\frac {6\,c\,x^{23/2}\,\left (b^2+a\,c\right )}{23} \]
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