Integrand size = 9, antiderivative size = 25 \[ \int \left (a+c x^2\right )^2 \, dx=a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \[ \int \left (a+c x^2\right )^2 \, dx=a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \]
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Rule 200
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx \\ & = a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (a+c x^2\right )^2 \, dx=a^2 x+\frac {2}{3} a c x^3+\frac {c^2 x^5}{5} \]
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Time = 2.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} x^{5} c^{2}\) | \(22\) |
default | \(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} x^{5} c^{2}\) | \(22\) |
norman | \(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} x^{5} c^{2}\) | \(22\) |
risch | \(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} x^{5} c^{2}\) | \(22\) |
parallelrisch | \(a^{2} x +\frac {2}{3} a c \,x^{3}+\frac {1}{5} x^{5} c^{2}\) | \(22\) |
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none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \left (a+c x^2\right )^2 \, dx=a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5} \]
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none
Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \]
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none
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, a c x^{3} + a^{2} x \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^2\right )^2 \, dx=a^2\,x+\frac {2\,a\,c\,x^3}{3}+\frac {c^2\,x^5}{5} \]
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