Integrand size = 18, antiderivative size = 89 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \]
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Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1122} \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{11} c x^{11} \left (a c+b^2\right )+\frac {1}{9} b x^9 \left (6 a c+b^2\right )+\frac {3}{7} a x^7 \left (a c+b^2\right )+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \]
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Rule 1122
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^2+3 a^2 b x^4+3 a \left (b^2+a c\right ) x^6+b \left (b^2+6 a c\right ) x^8+3 c \left (b^2+a c\right ) x^{10}+3 b c^2 x^{12}+c^3 x^{14}\right ) \, dx \\ & = \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \]
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Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {a^{3} x^{3}}{3}+\frac {3 a^{2} b \,x^{5}}{5}+\left (\frac {3}{7} c \,a^{2}+\frac {3}{7} b^{2} a \right ) x^{7}+\left (\frac {2}{3} a b c +\frac {1}{9} b^{3}\right ) x^{9}+\left (\frac {3}{11} a \,c^{2}+\frac {3}{11} b^{2} c \right ) x^{11}+\frac {3 b \,c^{2} x^{13}}{13}+\frac {c^{3} x^{15}}{15}\) | \(85\) |
gosper | \(\frac {1}{3} a^{3} x^{3}+\frac {3}{5} a^{2} b \,x^{5}+\frac {3}{7} x^{7} c \,a^{2}+\frac {3}{7} a \,b^{2} x^{7}+\frac {2}{3} x^{9} a b c +\frac {1}{9} b^{3} x^{9}+\frac {3}{11} x^{11} a \,c^{2}+\frac {3}{11} x^{11} b^{2} c +\frac {3}{13} b \,c^{2} x^{13}+\frac {1}{15} c^{3} x^{15}\) | \(88\) |
risch | \(\frac {1}{3} a^{3} x^{3}+\frac {3}{5} a^{2} b \,x^{5}+\frac {3}{7} x^{7} c \,a^{2}+\frac {3}{7} a \,b^{2} x^{7}+\frac {2}{3} x^{9} a b c +\frac {1}{9} b^{3} x^{9}+\frac {3}{11} x^{11} a \,c^{2}+\frac {3}{11} x^{11} b^{2} c +\frac {3}{13} b \,c^{2} x^{13}+\frac {1}{15} c^{3} x^{15}\) | \(88\) |
parallelrisch | \(\frac {1}{3} a^{3} x^{3}+\frac {3}{5} a^{2} b \,x^{5}+\frac {3}{7} x^{7} c \,a^{2}+\frac {3}{7} a \,b^{2} x^{7}+\frac {2}{3} x^{9} a b c +\frac {1}{9} b^{3} x^{9}+\frac {3}{11} x^{11} a \,c^{2}+\frac {3}{11} x^{11} b^{2} c +\frac {3}{13} b \,c^{2} x^{13}+\frac {1}{15} c^{3} x^{15}\) | \(88\) |
default | \(\frac {c^{3} x^{15}}{15}+\frac {3 b \,c^{2} x^{13}}{13}+\frac {\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{11}}{11}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{9}}{9}+\frac {\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{7}}{7}+\frac {3 a^{2} b \,x^{5}}{5}+\frac {a^{3} x^{3}}{3}\) | \(111\) |
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Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{7} + \frac {1}{3} \, a^{3} x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{5}}{5} + \frac {3 b c^{2} x^{13}}{13} + \frac {c^{3} x^{15}}{15} + x^{11} \cdot \left (\frac {3 a c^{2}}{11} + \frac {3 b^{2} c}{11}\right ) + x^{9} \cdot \left (\frac {2 a b c}{3} + \frac {b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c}{7} + \frac {3 a b^{2}}{7}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{7} + \frac {1}{3} \, a^{3} x^{3} \]
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Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, b^{2} c x^{11} + \frac {3}{11} \, a c^{2} x^{11} + \frac {1}{9} \, b^{3} x^{9} + \frac {2}{3} \, a b c x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{7} \, a^{2} c x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx=x^9\,\left (\frac {b^3}{9}+\frac {2\,a\,c\,b}{3}\right )+\frac {a^3\,x^3}{3}+\frac {c^3\,x^{15}}{15}+\frac {3\,a^2\,b\,x^5}{5}+\frac {3\,b\,c^2\,x^{13}}{13}+\frac {3\,a\,x^7\,\left (b^2+a\,c\right )}{7}+\frac {3\,c\,x^{11}\,\left (b^2+a\,c\right )}{11} \]
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