Integrand size = 20, antiderivative size = 118 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^3}{3 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{4 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \]
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Time = 0.09 (sec) , antiderivative size = 118, 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 = {78} \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=\frac {e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac {(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac {(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e)^2 (a+b x)^2}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^3}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^4}{b^3}+\frac {B e^2 (a+b x)^5}{b^3}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^2 (a+b x)^3}{3 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{4 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=a^2 A d^2 x+\frac {1}{2} a d (a B d+2 A (b d+a e)) x^2+\frac {1}{3} \left (2 a B d (b d+a e)+A \left (b^2 d^2+4 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (a^2 B e^2+2 a b e (2 B d+A e)+b^2 d (B d+2 A e)\right ) x^4+\frac {1}{5} b e (2 b B d+A b e+2 a B e) x^5+\frac {1}{6} b^2 B e^2 x^6 \]
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Time = 0.66 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {b^{2} B \,e^{2} x^{6}}{6}+\frac {\left (\left (b^{2} A +2 a b B \right ) e^{2}+2 b^{2} B d e \right ) x^{5}}{5}+\frac {\left (\left (2 a b A +a^{2} B \right ) e^{2}+2 \left (b^{2} A +2 a b B \right ) d e +b^{2} B \,d^{2}\right ) x^{4}}{4}+\frac {\left (a^{2} A \,e^{2}+2 \left (2 a b A +a^{2} B \right ) d e +\left (b^{2} A +2 a b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{2} A d e +\left (2 a b A +a^{2} B \right ) d^{2}\right ) x^{2}}{2}+a^{2} A \,d^{2} x\) | \(169\) |
norman | \(\frac {b^{2} B \,e^{2} x^{6}}{6}+\left (\frac {1}{5} A \,b^{2} e^{2}+\frac {2}{5} B a b \,e^{2}+\frac {2}{5} b^{2} B d e \right ) x^{5}+\left (\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{4} B \,a^{2} e^{2}+B a b d e +\frac {1}{4} b^{2} B \,d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} A \,e^{2}+\frac {4}{3} A a b d e +\frac {1}{3} A \,b^{2} d^{2}+\frac {2}{3} B \,a^{2} d e +\frac {2}{3} B a b \,d^{2}\right ) x^{3}+\left (a^{2} A d e +A a b \,d^{2}+\frac {1}{2} B \,a^{2} d^{2}\right ) x^{2}+a^{2} A \,d^{2} x\) | \(172\) |
gosper | \(\frac {1}{6} b^{2} B \,e^{2} x^{6}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} b^{2} B d e +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{4} x^{4} B \,a^{2} e^{2}+x^{4} B a b d e +\frac {1}{4} x^{4} b^{2} B \,d^{2}+\frac {1}{3} x^{3} a^{2} A \,e^{2}+\frac {4}{3} x^{3} A a b d e +\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} a^{2} A d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{2}+a^{2} A \,d^{2} x\) | \(200\) |
risch | \(\frac {1}{6} b^{2} B \,e^{2} x^{6}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} b^{2} B d e +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{4} x^{4} B \,a^{2} e^{2}+x^{4} B a b d e +\frac {1}{4} x^{4} b^{2} B \,d^{2}+\frac {1}{3} x^{3} a^{2} A \,e^{2}+\frac {4}{3} x^{3} A a b d e +\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} a^{2} A d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{2}+a^{2} A \,d^{2} x\) | \(200\) |
parallelrisch | \(\frac {1}{6} b^{2} B \,e^{2} x^{6}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} b^{2} B d e +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{4} x^{4} B \,a^{2} e^{2}+x^{4} B a b d e +\frac {1}{4} x^{4} b^{2} B \,d^{2}+\frac {1}{3} x^{3} a^{2} A \,e^{2}+\frac {4}{3} x^{3} A a b d e +\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} a^{2} A d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{2}+a^{2} A \,d^{2} x\) | \(200\) |
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Time = 0.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.42 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=\frac {1}{6} \, B b^{2} e^{2} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (2 \, B b^{2} d e + {\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.71 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=A a^{2} d^{2} x + \frac {B b^{2} e^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2} e^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {2 B b^{2} d e}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + \frac {A b^{2} d e}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{2}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.42 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=\frac {1}{6} \, B b^{2} e^{2} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (2 \, B b^{2} d e + {\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
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Time = 0.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.69 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=\frac {1}{6} \, B b^{2} e^{2} x^{6} + \frac {2}{5} \, B b^{2} d e x^{5} + \frac {2}{5} \, B a b e^{2} x^{5} + \frac {1}{5} \, A b^{2} e^{2} x^{5} + \frac {1}{4} \, B b^{2} d^{2} x^{4} + B a b d e x^{4} + \frac {1}{2} \, A b^{2} d e x^{4} + \frac {1}{4} \, B a^{2} e^{2} x^{4} + \frac {1}{2} \, A a b e^{2} x^{4} + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {2}{3} \, B a^{2} d e x^{3} + \frac {4}{3} \, A a b d e x^{3} + \frac {1}{3} \, A a^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + A a^{2} d e x^{2} + A a^{2} d^{2} x \]
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Time = 1.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx=x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^4\,\left (\frac {B\,a^2\,e^2}{4}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}\right )+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e\,x^5\,\left (A\,b\,e+2\,B\,a\,e+2\,B\,b\,d\right )}{5}+A\,a^2\,d^2\,x+\frac {B\,b^2\,e^2\,x^6}{6} \]
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