Integrand size = 27, antiderivative size = 77 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]
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Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {624, 45} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=-\frac {c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]
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Rule 45
Rule 624
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (c d^2+c d e x\right )^2 \left (a e^2+c d e x\right )^2 \, dx}{c^2 d^2 e^2} \\ & = \frac {\int \left (\left (c d^2-a e^2\right )^2 \left (c d^2+c d e x\right )^2-2 \left (c d^2-a e^2\right ) \left (c d^2+c d e x\right )^3+\left (c d^2+c d e x\right )^4\right ) \, dx}{c^2 d^2 e^2} \\ & = \frac {\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{30} x \left (10 a^2 e^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a c d e x \left (6 d^2+8 d e x+3 e^2 x^2\right )+c^2 d^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right ) \]
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Time = 2.38 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\frac {\left (e^{2} a +c \,d^{2}\right ) c d e \,x^{4}}{2}+\frac {\left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right ) x^{3}}{3}+a d e \left (e^{2} a +c \,d^{2}\right ) x^{2}+a^{2} d^{2} e^{2} x\) | \(93\) |
norman | \(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\left (\frac {1}{2} a c d \,e^{3}+\frac {1}{2} d^{3} e \,c^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} e^{4}+\frac {4}{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4}\right ) x^{3}+\left (a^{2} d \,e^{3}+a c \,d^{3} e \right ) x^{2}+a^{2} d^{2} e^{2} x\) | \(100\) |
risch | \(\frac {1}{5} c^{2} d^{2} e^{2} x^{5}+\frac {1}{2} a c d \,e^{3} x^{4}+\frac {1}{2} c^{2} d^{3} e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{4}+\frac {4}{3} x^{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4} x^{3}+a^{2} d \,e^{3} x^{2}+a c \,d^{3} e \,x^{2}+a^{2} d^{2} e^{2} x\) | \(106\) |
parallelrisch | \(\frac {1}{5} c^{2} d^{2} e^{2} x^{5}+\frac {1}{2} a c d \,e^{3} x^{4}+\frac {1}{2} c^{2} d^{3} e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{4}+\frac {4}{3} x^{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4} x^{3}+a^{2} d \,e^{3} x^{2}+a c \,d^{3} e \,x^{2}+a^{2} d^{2} e^{2} x\) | \(106\) |
gosper | \(\frac {x \left (6 c^{2} d^{2} e^{2} x^{4}+15 x^{3} a c d \,e^{3}+15 x^{3} d^{3} e \,c^{2}+10 x^{2} a^{2} e^{4}+40 x^{2} a c \,d^{2} e^{2}+10 x^{2} c^{2} d^{4}+30 a^{2} d \,e^{3} x +30 a c \,d^{3} e x +30 a^{2} d^{2} e^{2}\right )}{30}\) | \(107\) |
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Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.26 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + a^{2} d^{2} e^{2} x + \frac {1}{2} \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{3} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=a^{2} d^{2} e^{2} x + \frac {c^{2} d^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac {a c d e^{3}}{2} + \frac {c^{2} d^{3} e}{2}\right ) + x^{3} \left (\frac {a^{2} e^{4}}{3} + \frac {4 a c d^{2} e^{2}}{3} + \frac {c^{2} d^{4}}{3}\right ) + x^{2} \left (a^{2} d e^{3} + a c d^{3} e\right ) \]
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Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c d e x^{4} + a^{2} d^{2} e^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac {1}{3} \, {\left (2 \, c d e x^{3} + 3 \, {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a d e \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, c^{2} d^{3} e x^{4} + \frac {1}{2} \, a c d e^{3} x^{4} + \frac {1}{3} \, c^{2} d^{4} x^{3} + \frac {4}{3} \, a c d^{2} e^{2} x^{3} + \frac {1}{3} \, a^{2} e^{4} x^{3} + a c d^{3} e x^{2} + a^{2} d e^{3} x^{2} + a^{2} d^{2} e^{2} x \]
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Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=x^3\,\left (\frac {a^2\,e^4}{3}+\frac {4\,a\,c\,d^2\,e^2}{3}+\frac {c^2\,d^4}{3}\right )+x^2\,\left (a^2\,d\,e^3+c\,a\,d^3\,e\right )+x^4\,\left (\frac {c^2\,d^3\,e}{2}+\frac {a\,c\,d\,e^3}{2}\right )+a^2\,d^2\,e^2\,x+\frac {c^2\,d^2\,e^2\,x^5}{5} \]
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