Integrand size = 15, antiderivative size = 57 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^2}{e^2}-\frac {2 c d (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx \\ & = \frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=a d^2 x+a d e x^2+\frac {1}{3} \left (c d^2+a e^2\right ) x^3+\frac {1}{2} c d e x^4+\frac {1}{5} c e^2 x^5 \]
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Time = 2.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {c \,e^{2} x^{5}}{5}+\frac {x^{4} c d e}{2}+\frac {\left (e^{2} a +c \,d^{2}\right ) x^{3}}{3}+a d e \,x^{2}+x a \,d^{2}\) | \(48\) |
norman | \(\frac {c \,e^{2} x^{5}}{5}+\frac {x^{4} c d e}{2}+\left (\frac {e^{2} a}{3}+\frac {c \,d^{2}}{3}\right ) x^{3}+a d e \,x^{2}+x a \,d^{2}\) | \(49\) |
gosper | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\) | \(50\) |
risch | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\) | \(50\) |
parallelrisch | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\) | \(50\) |
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Time = 0.57 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + x^{3} \left (\frac {a e^{2}}{3} + \frac {c d^{2}}{3}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + a d^{2} x \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {c\,d^2}{3}+\frac {a\,e^2}{3}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x+a\,d\,e\,x^2+\frac {c\,d\,e\,x^4}{2} \]
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