Integrand size = 20, antiderivative size = 156 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{2 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^5}-\frac {c (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \]
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Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac {c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^4}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^5}{e^4}+\frac {c^2 (d+e x)^6}{e^4}\right ) \, dx \\ & = \frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{2 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^5}-\frac {c (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=a^2 d^2 x+a d (b d+a e) x^2+\frac {1}{3} \left (b^2 d^2+2 a c d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{2} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2+2 a c e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \]
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Time = 3.01 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {c^{2} e^{2} x^{7}}{7}+\frac {\left (2 b c \,e^{2}+2 d e \,c^{2}\right ) x^{6}}{6}+\frac {\left (c^{2} d^{2}+4 b c d e +e^{2} \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b c \,d^{2}+2 d e \left (2 a c +b^{2}\right )+2 a b \,e^{2}\right ) x^{4}}{4}+\frac {\left (d^{2} \left (2 a c +b^{2}\right )+4 a b d e +a^{2} e^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{2} d e +2 a b \,d^{2}\right ) x^{2}}{2}+a^{2} d^{2} x\) | \(155\) |
norman | \(\frac {c^{2} e^{2} x^{7}}{7}+\left (\frac {1}{3} b c \,e^{2}+\frac {1}{3} d e \,c^{2}\right ) x^{6}+\left (\frac {2}{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2}+\frac {4}{5} b c d e +\frac {1}{5} c^{2} d^{2}\right ) x^{5}+\left (\frac {1}{2} a b \,e^{2}+a c d e +\frac {1}{2} b^{2} d e +\frac {1}{2} b c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {4}{3} a b d e +\frac {2}{3} a c \,d^{2}+\frac {1}{3} b^{2} d^{2}\right ) x^{3}+\left (a^{2} d e +a b \,d^{2}\right ) x^{2}+a^{2} d^{2} x\) | \(156\) |
gosper | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2} x^{5}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} c^{2} d^{2} x^{5}+\frac {1}{2} x^{4} a b \,e^{2}+a c d e \,x^{4}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} a c \,d^{2} x^{3}+\frac {1}{3} d^{2} x^{3} b^{2}+a^{2} d e \,x^{2}+x^{2} a b \,d^{2}+a^{2} d^{2} x\) | \(179\) |
risch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2} x^{5}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} c^{2} d^{2} x^{5}+\frac {1}{2} x^{4} a b \,e^{2}+a c d e \,x^{4}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} a c \,d^{2} x^{3}+\frac {1}{3} d^{2} x^{3} b^{2}+a^{2} d e \,x^{2}+x^{2} a b \,d^{2}+a^{2} d^{2} x\) | \(179\) |
parallelrisch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2} x^{5}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} c^{2} d^{2} x^{5}+\frac {1}{2} x^{4} a b \,e^{2}+a c d e \,x^{4}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} a c \,d^{2} x^{3}+\frac {1}{3} d^{2} x^{3} b^{2}+a^{2} d e \,x^{2}+x^{2} a b \,d^{2}+a^{2} d^{2} x\) | \(179\) |
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Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a b d e + a^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=a^{2} d^{2} x + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \cdot \left (\frac {2 a c e^{2}}{5} + \frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {a b e^{2}}{2} + a c d e + \frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {4 a b d e}{3} + \frac {2 a c d^{2}}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a b d e + a^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
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Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + \frac {1}{3} \, b c e^{2} x^{6} + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {4}{5} \, b c d e x^{5} + \frac {1}{5} \, b^{2} e^{2} x^{5} + \frac {2}{5} \, a c e^{2} x^{5} + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{2} \, b^{2} d e x^{4} + a c d e x^{4} + \frac {1}{2} \, a b e^{2} x^{4} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {2}{3} \, a c d^{2} x^{3} + \frac {4}{3} \, a b d e x^{3} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a b d^{2} x^{2} + a^{2} d e x^{2} + a^{2} d^{2} x \]
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Time = 9.81 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {a^2\,e^2}{3}+\frac {4\,a\,b\,d\,e}{3}+\frac {2\,c\,a\,d^2}{3}+\frac {b^2\,d^2}{3}\right )+x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+x^4\,\left (\frac {b^2\,d\,e}{2}+\frac {c\,b\,d^2}{2}+\frac {a\,b\,e^2}{2}+a\,c\,d\,e\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a\,d\,x^2\,\left (a\,e+b\,d\right )+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \]
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