Integrand size = 20, antiderivative size = 118 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^4}{4 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \]
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Time = 0.11 (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} \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac {(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \]
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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)^3}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5}{b^3}+\frac {B e^2 (a+b x)^6}{b^3}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^2 (a+b x)^4}{4 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.90 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=a^3 A d^2 x+\frac {1}{2} a^2 d (3 A b d+a B d+2 a A e) x^2+\frac {1}{3} a \left (a B d (3 b d+2 a e)+A \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (a B \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )+A b \left (b^2 d^2+6 a b d e+3 a^2 e^2\right )\right ) x^4+\frac {1}{5} b \left (3 a^2 B e^2+3 a b e (2 B d+A e)+b^2 d (B d+2 A e)\right ) x^5+\frac {1}{6} b^2 e (2 b B d+A b e+3 a B e) x^6+\frac {1}{7} b^3 B e^2 x^7 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(110)=220\).
Time = 0.66 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.07
method | result | size |
default | \(\frac {b^{3} B \,e^{2} x^{7}}{7}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{2}+2 b^{3} B d e \right ) x^{6}}{6}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{2}+2 \left (b^{3} A +3 a \,b^{2} B \right ) d e +b^{3} B \,d^{2}\right ) x^{5}}{5}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{2}+2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{2}\right ) x^{4}}{4}+\frac {\left (a^{3} A \,e^{2}+2 \left (3 a^{2} b A +a^{3} B \right ) d e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} A d e +\left (3 a^{2} b A +a^{3} B \right ) d^{2}\right ) x^{2}}{2}+a^{3} A \,d^{2} x\) | \(244\) |
norman | \(\frac {b^{3} B \,e^{2} x^{7}}{7}+\left (\frac {1}{6} A \,b^{3} e^{2}+\frac {1}{2} B a \,b^{2} e^{2}+\frac {1}{3} b^{3} B d e \right ) x^{6}+\left (\frac {3}{5} A a \,b^{2} e^{2}+\frac {2}{5} A \,b^{3} d e +\frac {3}{5} B \,a^{2} b \,e^{2}+\frac {6}{5} B a \,b^{2} d e +\frac {1}{5} b^{3} B \,d^{2}\right ) x^{5}+\left (\frac {3}{4} A \,a^{2} b \,e^{2}+\frac {3}{2} A a \,b^{2} d e +\frac {1}{4} A \,b^{3} d^{2}+\frac {1}{4} B \,a^{3} e^{2}+\frac {3}{2} B \,a^{2} b d e +\frac {3}{4} B a \,b^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{3} A \,e^{2}+2 A \,a^{2} b d e +A a \,b^{2} d^{2}+\frac {2}{3} B \,a^{3} d e +B \,a^{2} b \,d^{2}\right ) x^{3}+\left (a^{3} A d e +\frac {3}{2} A \,a^{2} b \,d^{2}+\frac {1}{2} B \,a^{3} d^{2}\right ) x^{2}+a^{3} A \,d^{2} x\) | \(247\) |
gosper | \(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\) | \(288\) |
risch | \(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\) | \(288\) |
parallelrisch | \(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\) | \(288\) |
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Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (116) = 232\).
Time = 0.03 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.51 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=A a^{3} d^{2} x + \frac {B b^{3} e^{2} x^{7}}{7} + x^{6} \left (\frac {A b^{3} e^{2}}{6} + \frac {B a b^{2} e^{2}}{2} + \frac {B b^{3} d e}{3}\right ) + x^{5} \cdot \left (\frac {3 A a b^{2} e^{2}}{5} + \frac {2 A b^{3} d e}{5} + \frac {3 B a^{2} b e^{2}}{5} + \frac {6 B a b^{2} d e}{5} + \frac {B b^{3} d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b e^{2}}{4} + \frac {3 A a b^{2} d e}{2} + \frac {A b^{3} d^{2}}{4} + \frac {B a^{3} e^{2}}{4} + \frac {3 B a^{2} b d e}{2} + \frac {3 B a b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{3} e^{2}}{3} + 2 A a^{2} b d e + A a b^{2} d^{2} + \frac {2 B a^{3} d e}{3} + B a^{2} b d^{2}\right ) + x^{2} \left (A a^{3} d e + \frac {3 A a^{2} b d^{2}}{2} + \frac {B a^{3} d^{2}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (110) = 220\).
Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.43 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + \frac {1}{3} \, B b^{3} d e x^{6} + \frac {1}{2} \, B a b^{2} e^{2} x^{6} + \frac {1}{6} \, A b^{3} e^{2} x^{6} + \frac {1}{5} \, B b^{3} d^{2} x^{5} + \frac {6}{5} \, B a b^{2} d e x^{5} + \frac {2}{5} \, A b^{3} d e x^{5} + \frac {3}{5} \, B a^{2} b e^{2} x^{5} + \frac {3}{5} \, A a b^{2} e^{2} x^{5} + \frac {3}{4} \, B a b^{2} d^{2} x^{4} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {3}{2} \, B a^{2} b d e x^{4} + \frac {3}{2} \, A a b^{2} d e x^{4} + \frac {1}{4} \, B a^{3} e^{2} x^{4} + \frac {3}{4} \, A a^{2} b e^{2} x^{4} + B a^{2} b d^{2} x^{3} + A a b^{2} d^{2} x^{3} + \frac {2}{3} \, B a^{3} d e x^{3} + 2 \, A a^{2} b d e x^{3} + \frac {1}{3} \, A a^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{2} x^{2} + \frac {3}{2} \, A a^{2} b d^{2} x^{2} + A a^{3} d e x^{2} + A a^{3} d^{2} x \]
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Time = 1.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.96 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=x^4\,\left (\frac {B\,a^3\,e^2}{4}+\frac {3\,B\,a^2\,b\,d\,e}{2}+\frac {3\,A\,a^2\,b\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^2}{4}+\frac {3\,A\,a\,b^2\,d\,e}{2}+\frac {A\,b^3\,d^2}{4}\right )+x^3\,\left (\frac {2\,B\,a^3\,d\,e}{3}+\frac {A\,a^3\,e^2}{3}+B\,a^2\,b\,d^2+2\,A\,a^2\,b\,d\,e+A\,a\,b^2\,d^2\right )+x^5\,\left (\frac {3\,B\,a^2\,b\,e^2}{5}+\frac {6\,B\,a\,b^2\,d\,e}{5}+\frac {3\,A\,a\,b^2\,e^2}{5}+\frac {B\,b^3\,d^2}{5}+\frac {2\,A\,b^3\,d\,e}{5}\right )+A\,a^3\,d^2\,x+\frac {a^2\,d\,x^2\,\left (2\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e\,x^6\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{6}+\frac {B\,b^3\,e^2\,x^7}{7} \]
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