Integrand size = 18, antiderivative size = 77 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^3}{3 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^4}{4 e^3}+\frac {b B (d+e x)^5}{5 e^3} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=-\frac {(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac {(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac {b B (d+e x)^5}{5 e^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^2}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^3}{e^2}+\frac {b B (d+e x)^4}{e^2}\right ) \, dx \\ & = \frac {(b d-a e) (B d-A e) (d+e x)^3}{3 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^4}{4 e^3}+\frac {b B (d+e x)^5}{5 e^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=a A d^2 x+\frac {1}{2} d (A b d+a B d+2 a A e) x^2+\frac {1}{3} \left (b B d^2+2 A b d e+2 a B d e+a A e^2\right ) x^3+\frac {1}{4} e (2 b B d+A b e+a B e) x^4+\frac {1}{5} b B e^2 x^5 \]
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Time = 0.65 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {b B \,e^{2} x^{5}}{5}+\frac {\left (\left (A b +B a \right ) e^{2}+2 b B d e \right ) x^{4}}{4}+\frac {\left (A a \,e^{2}+2 \left (A b +B a \right ) d e +b B \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 A a d e +\left (A b +B a \right ) d^{2}\right ) x^{2}}{2}+A a \,d^{2} x\) | \(94\) |
norman | \(\frac {b B \,e^{2} x^{5}}{5}+\left (\frac {1}{4} A b \,e^{2}+\frac {1}{4} B a \,e^{2}+\frac {1}{2} b B d e \right ) x^{4}+\left (\frac {1}{3} A a \,e^{2}+\frac {2}{3} A b d e +\frac {2}{3} B a d e +\frac {1}{3} b B \,d^{2}\right ) x^{3}+\left (A a d e +\frac {1}{2} A b \,d^{2}+\frac {1}{2} B a \,d^{2}\right ) x^{2}+A a \,d^{2} x\) | \(99\) |
gosper | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
risch | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
parallelrisch | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
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Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=A a d^{2} x + \frac {B b e^{2} x^{5}}{5} + x^{4} \left (\frac {A b e^{2}}{4} + \frac {B a e^{2}}{4} + \frac {B b d e}{2}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {2 A b d e}{3} + \frac {2 B a d e}{3} + \frac {B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {A b d^{2}}{2} + \frac {B a d^{2}}{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + \frac {1}{2} \, B b d e x^{4} + \frac {1}{4} \, B a e^{2} x^{4} + \frac {1}{4} \, A b e^{2} x^{4} + \frac {1}{3} \, B b d^{2} x^{3} + \frac {2}{3} \, B a d e x^{3} + \frac {2}{3} \, A b d e x^{3} + \frac {1}{3} \, A a e^{2} x^{3} + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{2} \, A b d^{2} x^{2} + A a d e x^{2} + A a d^{2} x \]
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Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^2\,\left (\frac {A\,b\,d^2}{2}+\frac {B\,a\,d^2}{2}+A\,a\,d\,e\right )+x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,a\,e^2}{4}+\frac {B\,b\,d\,e}{2}\right )+A\,a\,d^2\,x+\frac {B\,b\,e^2\,x^5}{5} \]
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