Integrand size = 16, antiderivative size = 54 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \]
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Time = 0.02 (sec) , antiderivative size = 54, 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 = {1608, 1122} \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^3}{3}+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \]
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Rule 1122
Rule 1608
Rubi steps \begin{align*} \text {integral}& = \int x^2 \left (a+b x^2+c x^4\right )^2 \, dx \\ & = \int \left (a^2 x^2+2 a b x^4+\left (b^2+2 a c\right ) x^6+2 b c x^8+c^2 x^{10}\right ) \, dx \\ & = \frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \]
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Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {a^{2} x^{3}}{3}+\frac {2 a b \,x^{5}}{5}+\frac {\left (2 a c +b^{2}\right ) x^{7}}{7}+\frac {2 b c \,x^{9}}{9}+\frac {c^{2} x^{11}}{11}\) | \(45\) |
norman | \(\frac {c^{2} x^{11}}{11}+\frac {2 b c \,x^{9}}{9}+\left (\frac {2 a c}{7}+\frac {b^{2}}{7}\right ) x^{7}+\frac {2 a b \,x^{5}}{5}+\frac {a^{2} x^{3}}{3}\) | \(46\) |
risch | \(\frac {1}{3} a^{2} x^{3}+\frac {2}{5} a b \,x^{5}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {2}{9} b c \,x^{9}+\frac {1}{11} c^{2} x^{11}\) | \(47\) |
parallelrisch | \(\frac {1}{3} a^{2} x^{3}+\frac {2}{5} a b \,x^{5}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {2}{9} b c \,x^{9}+\frac {1}{11} c^{2} x^{11}\) | \(47\) |
gosper | \(\frac {x^{3} \left (315 c^{2} x^{8}+770 b c \,x^{6}+990 a c \,x^{4}+495 b^{2} x^{4}+1386 a b \,x^{2}+1155 a^{2}\right )}{3465}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {2}{9} \, b c x^{9} + \frac {1}{7} \, {\left (b^{2} + 2 \, a c\right )} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {2 b c x^{9}}{9} + \frac {c^{2} x^{11}}{11} + x^{7} \cdot \left (\frac {2 a c}{7} + \frac {b^{2}}{7}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {2}{9} \, b c x^{9} + \frac {1}{7} \, b^{2} x^{7} + \frac {1}{3} \, a^{2} x^{3} + \frac {2}{35} \, {\left (5 \, c x^{7} + 7 \, b x^{5}\right )} a \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{11} \, c^{2} x^{11} + \frac {2}{9} \, b c x^{9} + \frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \left (a x+b x^3+c x^5\right )^2 \, dx=x^7\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {a^2\,x^3}{3}+\frac {c^2\,x^{11}}{11}+\frac {2\,a\,b\,x^5}{5}+\frac {2\,b\,c\,x^9}{9} \]
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