Integrand size = 22, antiderivative size = 54 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1599, 712} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]
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Rule 712
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int x^2 \left (a+b x+c x^2\right )^2 \, dx \\ & = \int \left (a^2 x^2+2 a b x^3+\left (b^2+2 a c\right ) x^4+2 b c x^5+c^2 x^6\right ) \, dx \\ & = \frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]
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Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {a^{2} x^{3}}{3}+\frac {a b \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) x^{5}}{5}+\frac {b c \,x^{6}}{3}+\frac {c^{2} x^{7}}{7}\) | \(45\) |
gosper | \(\frac {x^{3} \left (30 c^{2} x^{4}+70 b c \,x^{3}+84 a c \,x^{2}+42 b^{2} x^{2}+105 a b x +70 a^{2}\right )}{210}\) | \(47\) |
risch | \(\frac {1}{3} a^{2} x^{3}+\frac {1}{2} a b \,x^{4}+\frac {2}{5} x^{5} a c +\frac {1}{5} b^{2} x^{5}+\frac {1}{3} b c \,x^{6}+\frac {1}{7} c^{2} x^{7}\) | \(47\) |
parallelrisch | \(\frac {1}{3} a^{2} x^{3}+\frac {1}{2} a b \,x^{4}+\frac {2}{5} x^{5} a c +\frac {1}{5} b^{2} x^{5}+\frac {1}{3} b c \,x^{6}+\frac {1}{7} c^{2} x^{7}\) | \(47\) |
norman | \(\frac {\left (\frac {2 a c}{5}+\frac {b^{2}}{5}\right ) x^{6}+\frac {a^{2} x^{4}}{3}+\frac {c^{2} x^{8}}{7}+\frac {a b \,x^{5}}{2}+\frac {b c \,x^{7}}{3}}{x}\) | \(50\) |
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {a^{2} x^{3}}{3} + \frac {a b x^{4}}{2} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a c}{5} + \frac {b^{2}}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \]
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=\frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{5} \, b^{2} x^{5} + \frac {2}{5} \, a c x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{3} \, a^{2} x^{3} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx=x^5\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+\frac {a^2\,x^3}{3}+\frac {c^2\,x^7}{7}+\frac {a\,b\,x^4}{2}+\frac {b\,c\,x^6}{3} \]
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