Integrand size = 46, antiderivative size = 193 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^5}+\frac {(b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^4}{2 b^5}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{5 b^5}+\frac {d f (b d e+b c f-2 a d f) (a+b x)^6}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5} \]
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Time = 0.17 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2084, 90} \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac {d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac {(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac {(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5} \]
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Rule 90
Rule 2084
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^2 (c+d x)^2 (e+f x)^2 \, dx \\ & = \int \left (\frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^2}{b^4}+\frac {2 (b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^3}{b^4}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^4}{b^4}+\frac {2 d f (b d e+b c f-2 a d f) (a+b x)^5}{b^4}+\frac {d^2 f^2 (a+b x)^6}{b^4}\right ) \, dx \\ & = \frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^5}+\frac {(b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^4}{2 b^5}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{5 b^5}+\frac {d f (b d e+b c f-2 a d f) (a+b x)^6}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.25 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=a^2 c^2 e^2 x+a c e (b c e+a d e+a c f) x^2+\frac {1}{3} \left (b^2 c^2 e^2+4 a b c e (d e+c f)+a^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^3+\frac {1}{2} \left (b^2 c e (d e+c f)+a^2 d f (d e+c f)+a b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^4+\frac {1}{5} \left (a^2 d^2 f^2+4 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{3} b d f (b d e+b c f+a d f) x^6+\frac {1}{7} b^2 d^2 f^2 x^7 \]
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Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {d^{2} f^{2} b^{2} x^{7}}{7}+\frac {\left (a d f +f b c +b d e \right ) b d f \,x^{6}}{3}+\frac {\left (2 \left (a c f +e d a +e b c \right ) b d f +\left (a d f +f b c +b d e \right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a c e b d f +2 \left (a c f +e d a +e b c \right ) \left (a d f +f b c +b d e \right )\right ) x^{4}}{4}+\frac {\left (2 a c e \left (a d f +f b c +b d e \right )+\left (a c f +e d a +e b c \right )^{2}\right ) x^{3}}{3}+a c e \left (a c f +e d a +e b c \right ) x^{2}+a^{2} c^{2} e^{2} x\) | \(188\) |
norman | \(\frac {d^{2} f^{2} b^{2} x^{7}}{7}+\left (\frac {1}{3} d^{2} f^{2} a b +\frac {1}{3} b^{2} c d \,f^{2}+\frac {1}{3} b^{2} d^{2} e f \right ) x^{6}+\left (\frac {1}{5} a^{2} d^{2} f^{2}+\frac {4}{5} a b c d \,f^{2}+\frac {4}{5} a b \,d^{2} e f +\frac {1}{5} b^{2} c^{2} f^{2}+\frac {4}{5} b^{2} c d e f +\frac {1}{5} b^{2} d^{2} e^{2}\right ) x^{5}+\left (\frac {1}{2} a^{2} c d \,f^{2}+\frac {1}{2} a^{2} d^{2} e f +\frac {1}{2} a b \,c^{2} f^{2}+2 a c e b d f +\frac {1}{2} a b \,d^{2} e^{2}+\frac {1}{2} b^{2} c^{2} e f +\frac {1}{2} b^{2} c d \,e^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} c^{2} f^{2}+\frac {4}{3} a^{2} c d e f +\frac {1}{3} a^{2} d^{2} e^{2}+\frac {4}{3} a b \,c^{2} e f +\frac {4}{3} a b c d \,e^{2}+\frac {1}{3} b^{2} c^{2} e^{2}\right ) x^{3}+\left (a^{2} c^{2} e f +a^{2} c d \,e^{2}+e^{2} c^{2} a b \right ) x^{2}+a^{2} c^{2} e^{2} x\) | \(297\) |
gosper | \(\frac {x \left (30 d^{2} f^{2} b^{2} x^{6}+70 x^{5} d^{2} f^{2} a b +70 x^{5} b^{2} c d \,f^{2}+70 x^{5} b^{2} d^{2} e f +42 x^{4} a^{2} d^{2} f^{2}+168 x^{4} a b c d \,f^{2}+168 x^{4} a b \,d^{2} e f +42 x^{4} b^{2} c^{2} f^{2}+168 x^{4} b^{2} c d e f +42 x^{4} b^{2} d^{2} e^{2}+105 x^{3} a^{2} c d \,f^{2}+105 x^{3} a^{2} d^{2} e f +105 x^{3} a b \,c^{2} f^{2}+420 x^{3} a c e b d f +105 x^{3} a b \,d^{2} e^{2}+105 x^{3} b^{2} c^{2} e f +105 x^{3} b^{2} c d \,e^{2}+70 x^{2} a^{2} c^{2} f^{2}+280 x^{2} a^{2} c d e f +70 x^{2} a^{2} d^{2} e^{2}+280 x^{2} a b \,c^{2} e f +280 x^{2} a b c d \,e^{2}+70 x^{2} b^{2} c^{2} e^{2}+210 a^{2} c^{2} e f x +210 a^{2} c d \,e^{2} x +210 a b \,c^{2} e^{2} x +210 a^{2} c^{2} e^{2}\right )}{210}\) | \(347\) |
risch | \(\frac {4}{5} x^{5} a b c d \,f^{2}+\frac {4}{5} x^{5} a b \,d^{2} e f +\frac {4}{5} x^{5} b^{2} c d e f +\frac {4}{3} x^{3} a^{2} c d e f +\frac {4}{3} x^{3} a b \,c^{2} e f +\frac {4}{3} x^{3} a b c d \,e^{2}+a^{2} c^{2} e^{2} x +\frac {1}{7} d^{2} f^{2} b^{2} x^{7}+2 x^{4} a c e b d f +\frac {1}{2} x^{4} a^{2} d^{2} e f +a^{2} c^{2} e f \,x^{2}+\frac {1}{2} x^{4} a b \,c^{2} f^{2}+\frac {1}{2} x^{4} a b \,d^{2} e^{2}+\frac {1}{2} x^{4} b^{2} c^{2} e f +\frac {1}{2} x^{4} b^{2} c d \,e^{2}+a^{2} c d \,e^{2} x^{2}+a b \,c^{2} e^{2} x^{2}+\frac {1}{3} a b \,d^{2} f^{2} x^{6}+\frac {1}{3} b^{2} c d \,f^{2} x^{6}+\frac {1}{3} b^{2} d^{2} e f \,x^{6}+\frac {1}{2} x^{4} a^{2} c d \,f^{2}+\frac {1}{5} x^{5} a^{2} d^{2} f^{2}+\frac {1}{5} x^{5} b^{2} c^{2} f^{2}+\frac {1}{5} x^{5} b^{2} d^{2} e^{2}+\frac {1}{3} x^{3} a^{2} c^{2} f^{2}+\frac {1}{3} x^{3} a^{2} d^{2} e^{2}+\frac {1}{3} x^{3} b^{2} c^{2} e^{2}\) | \(347\) |
parallelrisch | \(\frac {4}{5} x^{5} a b c d \,f^{2}+\frac {4}{5} x^{5} a b \,d^{2} e f +\frac {4}{5} x^{5} b^{2} c d e f +\frac {4}{3} x^{3} a^{2} c d e f +\frac {4}{3} x^{3} a b \,c^{2} e f +\frac {4}{3} x^{3} a b c d \,e^{2}+a^{2} c^{2} e^{2} x +\frac {1}{7} d^{2} f^{2} b^{2} x^{7}+2 x^{4} a c e b d f +\frac {1}{2} x^{4} a^{2} d^{2} e f +a^{2} c^{2} e f \,x^{2}+\frac {1}{2} x^{4} a b \,c^{2} f^{2}+\frac {1}{2} x^{4} a b \,d^{2} e^{2}+\frac {1}{2} x^{4} b^{2} c^{2} e f +\frac {1}{2} x^{4} b^{2} c d \,e^{2}+a^{2} c d \,e^{2} x^{2}+a b \,c^{2} e^{2} x^{2}+\frac {1}{3} a b \,d^{2} f^{2} x^{6}+\frac {1}{3} b^{2} c d \,f^{2} x^{6}+\frac {1}{3} b^{2} d^{2} e f \,x^{6}+\frac {1}{2} x^{4} a^{2} c d \,f^{2}+\frac {1}{5} x^{5} a^{2} d^{2} f^{2}+\frac {1}{5} x^{5} b^{2} c^{2} f^{2}+\frac {1}{5} x^{5} b^{2} d^{2} e^{2}+\frac {1}{3} x^{3} a^{2} c^{2} f^{2}+\frac {1}{3} x^{3} a^{2} d^{2} e^{2}+\frac {1}{3} x^{3} b^{2} c^{2} e^{2}\) | \(347\) |
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Time = 0.23 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.39 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + a^{2} c^{2} e^{2} x + \frac {1}{3} \, {\left (b^{2} d^{2} e f + {\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{5} + \frac {1}{2} \, {\left ({\left (b^{2} c d + a b d^{2}\right )} e^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f + {\left (a b c^{2} + a^{2} c d\right )} f^{2}\right )} x^{4} + \frac {1}{3} \, {\left (a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} e f\right )} x^{3} + {\left (a^{2} c^{2} e f + {\left (a b c^{2} + a^{2} c d\right )} e^{2}\right )} x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.79 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=a^{2} c^{2} e^{2} x + \frac {b^{2} d^{2} f^{2} x^{7}}{7} + x^{6} \left (\frac {a b d^{2} f^{2}}{3} + \frac {b^{2} c d f^{2}}{3} + \frac {b^{2} d^{2} e f}{3}\right ) + x^{5} \left (\frac {a^{2} d^{2} f^{2}}{5} + \frac {4 a b c d f^{2}}{5} + \frac {4 a b d^{2} e f}{5} + \frac {b^{2} c^{2} f^{2}}{5} + \frac {4 b^{2} c d e f}{5} + \frac {b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} c d f^{2}}{2} + \frac {a^{2} d^{2} e f}{2} + \frac {a b c^{2} f^{2}}{2} + 2 a b c d e f + \frac {a b d^{2} e^{2}}{2} + \frac {b^{2} c^{2} e f}{2} + \frac {b^{2} c d e^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} c^{2} f^{2}}{3} + \frac {4 a^{2} c d e f}{3} + \frac {a^{2} d^{2} e^{2}}{3} + \frac {4 a b c^{2} e f}{3} + \frac {4 a b c d e^{2}}{3} + \frac {b^{2} c^{2} e^{2}}{3}\right ) + x^{2} \left (a^{2} c^{2} e f + a^{2} c d e^{2} + a b c^{2} e^{2}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.93 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} b d f x^{6} + a^{2} c^{2} e^{2} x + \frac {1}{5} \, {\left (b d e + b c f + a d f\right )}^{2} x^{5} + \frac {1}{3} \, {\left (b c e + a d e + a c f\right )}^{2} x^{3} + \frac {1}{6} \, {\left (3 \, b d f x^{4} + 4 \, {\left (b d e + b c f + a d f\right )} x^{3} + 6 \, {\left (b c e + a d e + a c f\right )} x^{2}\right )} a c e + \frac {1}{10} \, {\left (4 \, b d f x^{5} + 5 \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{4}\right )} {\left (b c e + a d e + a c f\right )} \]
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Time = 0.35 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.79 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} e f x^{6} + \frac {1}{3} \, b^{2} c d f^{2} x^{6} + \frac {1}{3} \, a b d^{2} f^{2} x^{6} + \frac {1}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, b^{2} c d e f x^{5} + \frac {4}{5} \, a b d^{2} e f x^{5} + \frac {1}{5} \, b^{2} c^{2} f^{2} x^{5} + \frac {4}{5} \, a b c d f^{2} x^{5} + \frac {1}{5} \, a^{2} d^{2} f^{2} x^{5} + \frac {1}{2} \, b^{2} c d e^{2} x^{4} + \frac {1}{2} \, a b d^{2} e^{2} x^{4} + \frac {1}{2} \, b^{2} c^{2} e f x^{4} + 2 \, a b c d e f x^{4} + \frac {1}{2} \, a^{2} d^{2} e f x^{4} + \frac {1}{2} \, a b c^{2} f^{2} x^{4} + \frac {1}{2} \, a^{2} c d f^{2} x^{4} + \frac {1}{3} \, b^{2} c^{2} e^{2} x^{3} + \frac {4}{3} \, a b c d e^{2} x^{3} + \frac {1}{3} \, a^{2} d^{2} e^{2} x^{3} + \frac {4}{3} \, a b c^{2} e f x^{3} + \frac {4}{3} \, a^{2} c d e f x^{3} + \frac {1}{3} \, a^{2} c^{2} f^{2} x^{3} + a b c^{2} e^{2} x^{2} + a^{2} c d e^{2} x^{2} + a^{2} c^{2} e f x^{2} + a^{2} c^{2} e^{2} x \]
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Time = 0.05 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.40 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=x^4\,\left (\frac {a^2\,c\,d\,f^2}{2}+\frac {a^2\,d^2\,e\,f}{2}+\frac {a\,b\,c^2\,f^2}{2}+2\,a\,b\,c\,d\,e\,f+\frac {a\,b\,d^2\,e^2}{2}+\frac {b^2\,c^2\,e\,f}{2}+\frac {b^2\,c\,d\,e^2}{2}\right )+x^3\,\left (\frac {a^2\,c^2\,f^2}{3}+\frac {4\,a^2\,c\,d\,e\,f}{3}+\frac {a^2\,d^2\,e^2}{3}+\frac {4\,a\,b\,c^2\,e\,f}{3}+\frac {4\,a\,b\,c\,d\,e^2}{3}+\frac {b^2\,c^2\,e^2}{3}\right )+x^5\,\left (\frac {a^2\,d^2\,f^2}{5}+\frac {4\,a\,b\,c\,d\,f^2}{5}+\frac {4\,a\,b\,d^2\,e\,f}{5}+\frac {b^2\,c^2\,f^2}{5}+\frac {4\,b^2\,c\,d\,e\,f}{5}+\frac {b^2\,d^2\,e^2}{5}\right )+a^2\,c^2\,e^2\,x+\frac {b^2\,d^2\,f^2\,x^7}{7}+a\,c\,e\,x^2\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )+\frac {b\,d\,f\,x^6\,\left (a\,d\,f+b\,c\,f+b\,d\,e\right )}{3} \]
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