Integrand size = 16, antiderivative size = 89 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 x^2}{2}+\frac {3}{4} a^2 b x^4+\frac {1}{2} a \left (b^2+a c\right ) x^6+\frac {1}{8} b \left (b^2+6 a c\right ) x^8+\frac {3}{10} c \left (b^2+a c\right ) x^{10}+\frac {1}{4} b c^2 x^{12}+\frac {c^3 x^{14}}{14} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1121, 625} \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 x^2}{2}+\frac {3}{4} a^2 b x^4+\frac {3}{10} c x^{10} \left (a c+b^2\right )+\frac {1}{8} b x^8 \left (6 a c+b^2\right )+\frac {1}{2} a x^6 \left (a c+b^2\right )+\frac {1}{4} b c^2 x^{12}+\frac {c^3 x^{14}}{14} \]
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Rule 625
Rule 1121
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \left (a+b x+c x^2\right )^3 \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a^3+3 a^2 b x+3 a b^2 \left (1+\frac {a c}{b^2}\right ) x^2+b^3 \left (1+\frac {6 a c}{b^2}\right ) x^3+3 b^2 c \left (1+\frac {a c}{b^2}\right ) x^4+3 b c^2 x^5+c^3 x^6\right ) \, dx,x,x^2\right ) \\ & = \frac {a^3 x^2}{2}+\frac {3}{4} a^2 b x^4+\frac {1}{2} a \left (b^2+a c\right ) x^6+\frac {1}{8} b \left (b^2+6 a c\right ) x^8+\frac {3}{10} c \left (b^2+a c\right ) x^{10}+\frac {1}{4} b c^2 x^{12}+\frac {c^3 x^{14}}{14} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{280} x^2 \left (140 a^3+210 a^2 b x^2+140 a \left (b^2+a c\right ) x^4+35 b \left (b^2+6 a c\right ) x^6+84 c \left (b^2+a c\right ) x^8+70 b c^2 x^{10}+20 c^3 x^{12}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {a^{3} x^{2}}{2}+\frac {3 a^{2} b \,x^{4}}{4}+\left (\frac {1}{2} c \,a^{2}+\frac {1}{2} b^{2} a \right ) x^{6}+\left (\frac {3}{4} a b c +\frac {1}{8} b^{3}\right ) x^{8}+\left (\frac {3}{10} a \,c^{2}+\frac {3}{10} b^{2} c \right ) x^{10}+\frac {b \,c^{2} x^{12}}{4}+\frac {c^{3} x^{14}}{14}\) | \(85\) |
gosper | \(\frac {1}{2} a^{3} x^{2}+\frac {3}{4} a^{2} b \,x^{4}+\frac {1}{2} x^{6} c \,a^{2}+\frac {1}{2} b^{2} x^{6} a +\frac {3}{4} x^{8} a b c +\frac {1}{8} b^{3} x^{8}+\frac {3}{10} x^{10} a \,c^{2}+\frac {3}{10} x^{10} b^{2} c +\frac {1}{4} b \,c^{2} x^{12}+\frac {1}{14} c^{3} x^{14}\) | \(88\) |
risch | \(\frac {1}{2} a^{3} x^{2}+\frac {3}{4} a^{2} b \,x^{4}+\frac {1}{2} x^{6} c \,a^{2}+\frac {1}{2} b^{2} x^{6} a +\frac {3}{4} x^{8} a b c +\frac {1}{8} b^{3} x^{8}+\frac {3}{10} x^{10} a \,c^{2}+\frac {3}{10} x^{10} b^{2} c +\frac {1}{4} b \,c^{2} x^{12}+\frac {1}{14} c^{3} x^{14}\) | \(88\) |
parallelrisch | \(\frac {1}{2} a^{3} x^{2}+\frac {3}{4} a^{2} b \,x^{4}+\frac {1}{2} x^{6} c \,a^{2}+\frac {1}{2} b^{2} x^{6} a +\frac {3}{4} x^{8} a b c +\frac {1}{8} b^{3} x^{8}+\frac {3}{10} x^{10} a \,c^{2}+\frac {3}{10} x^{10} b^{2} c +\frac {1}{4} b \,c^{2} x^{12}+\frac {1}{14} c^{3} x^{14}\) | \(88\) |
default | \(\frac {c^{3} x^{14}}{14}+\frac {b \,c^{2} x^{12}}{4}+\frac {\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{10}}{10}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{6}}{6}+\frac {3 a^{2} b \,x^{4}}{4}+\frac {a^{3} x^{2}}{2}\) | \(111\) |
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Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{14} \, c^{3} x^{14} + \frac {1}{4} \, b c^{2} x^{12} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{6} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{4}}{4} + \frac {b c^{2} x^{12}}{4} + \frac {c^{3} x^{14}}{14} + x^{10} \cdot \left (\frac {3 a c^{2}}{10} + \frac {3 b^{2} c}{10}\right ) + x^{8} \cdot \left (\frac {3 a b c}{4} + \frac {b^{3}}{8}\right ) + x^{6} \left (\frac {a^{2} c}{2} + \frac {a b^{2}}{2}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{14} \, c^{3} x^{14} + \frac {1}{4} \, b c^{2} x^{12} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{6} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{14} \, c^{3} x^{14} + \frac {1}{4} \, b c^{2} x^{12} + \frac {3}{10} \, b^{2} c x^{10} + \frac {3}{10} \, a c^{2} x^{10} + \frac {1}{8} \, b^{3} x^{8} + \frac {3}{4} \, a b c x^{8} + \frac {1}{2} \, a b^{2} x^{6} + \frac {1}{2} \, a^{2} c x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, a^{3} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^2+c x^4\right )^3 \, dx=x^8\,\left (\frac {b^3}{8}+\frac {3\,a\,c\,b}{4}\right )+\frac {a^3\,x^2}{2}+\frac {c^3\,x^{14}}{14}+\frac {3\,a^2\,b\,x^4}{4}+\frac {b\,c^2\,x^{12}}{4}+\frac {a\,x^6\,\left (b^2+a\,c\right )}{2}+\frac {3\,c\,x^{10}\,\left (b^2+a\,c\right )}{10} \]
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