Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^3}{3 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{4 b^3}+\frac {B e (a+b x)^5}{5 b^3} \]
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Time = 0.06 (sec) , antiderivative size = 75, 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} \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac {(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {B e (a+b x)^5}{5 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e) (a+b x)^2}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^3}{b^2}+\frac {B e (a+b x)^4}{b^2}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e) (a+b x)^3}{3 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{4 b^3}+\frac {B e (a+b x)^5}{5 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.28 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=a^2 A d x+\frac {1}{2} a (2 A b d+a B d+a A e) x^2+\frac {1}{3} \left (A b^2 d+2 a b B d+2 a A b e+a^2 B e\right ) x^3+\frac {1}{4} b (b B d+A b e+2 a B e) x^4+\frac {1}{5} b^2 B e x^5 \]
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Time = 0.66 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.32
method | result | size |
norman | \(\frac {b^{2} B e \,x^{5}}{5}+\left (\frac {1}{4} A \,b^{2} e +\frac {1}{2} B a b e +\frac {1}{4} b^{2} B d \right ) x^{4}+\left (\frac {2}{3} A a b e +\frac {1}{3} A \,b^{2} d +\frac {1}{3} B \,a^{2} e +\frac {2}{3} B a b d \right ) x^{3}+\left (\frac {1}{2} a^{2} A e +A a b d +\frac {1}{2} B \,a^{2} d \right ) x^{2}+a^{2} A d x\) | \(99\) |
default | \(\frac {b^{2} B e \,x^{5}}{5}+\frac {\left (\left (b^{2} A +2 a b B \right ) e +b^{2} B d \right ) x^{4}}{4}+\frac {\left (\left (2 a b A +a^{2} B \right ) e +\left (b^{2} A +2 a b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{2} A e +\left (2 a b A +a^{2} B \right ) d \right ) x^{2}}{2}+a^{2} A d x\) | \(101\) |
gosper | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
risch | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
parallelrisch | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
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Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2}\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.55 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=A a^{2} d x + \frac {B b^{2} e x^{5}}{5} + x^{4} \left (\frac {A b^{2} e}{4} + \frac {B a b e}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {2 A a b e}{3} + \frac {A b^{2} d}{3} + \frac {B a^{2} e}{3} + \frac {2 B a b d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + A a b d + \frac {B a^{2} d}{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2}\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a b e x^{4} + \frac {1}{4} \, A b^{2} e x^{4} + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{3} \, B a^{2} e x^{3} + \frac {2}{3} \, A a b e x^{3} + \frac {1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac {1}{2} \, A a^{2} e x^{2} + A a^{2} d x \]
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Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.31 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {B\,a\,b\,e}{2}\right )+A\,a^2\,d\,x+\frac {B\,b^2\,e\,x^5}{5} \]
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