Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^7}{7 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^8}{8 b^3}+\frac {B e (a+b x)^9}{9 b^3} \]
[Out]
Time = 0.14 (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} \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {(a+b x)^8 (-2 a B e+A b e+b B d)}{8 b^3}+\frac {(a+b x)^7 (A b-a B) (b d-a e)}{7 b^3}+\frac {B e (a+b x)^9}{9 b^3} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e) (a+b x)^6}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^7}{b^2}+\frac {B e (a+b x)^8}{b^2}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e) (a+b x)^7}{7 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^8}{8 b^3}+\frac {B e (a+b x)^9}{9 b^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(231\) vs. \(2(75)=150\).
Time = 0.09 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.08 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {1}{504} x \left (84 a^6 (3 A (2 d+e x)+B x (3 d+2 e x))+126 a^4 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+252 a^5 b x (B x (4 d+3 e x)+A (6 d+4 e x))+168 a^3 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+36 a^2 b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+18 a b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+b^6 x^6 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(278\) vs. \(2(69)=138\).
Time = 0.71 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.72
method | result | size |
norman | \(\frac {b^{6} B e \,x^{9}}{9}+\left (\frac {1}{8} A \,b^{6} e +\frac {3}{4} B a \,b^{5} e +\frac {1}{8} b^{6} B d \right ) x^{8}+\left (\frac {6}{7} A a \,b^{5} e +\frac {1}{7} A \,b^{6} d +\frac {15}{7} B \,a^{2} b^{4} e +\frac {6}{7} B a \,b^{5} d \right ) x^{7}+\left (\frac {5}{2} A \,a^{2} b^{4} e +A a \,b^{5} d +\frac {10}{3} B \,a^{3} b^{3} e +\frac {5}{2} B \,a^{2} b^{4} d \right ) x^{6}+\left (4 A \,a^{3} b^{3} e +3 A \,a^{2} b^{4} d +3 B \,a^{4} b^{2} e +4 B \,a^{3} b^{3} d \right ) x^{5}+\left (\frac {15}{4} A \,a^{4} b^{2} e +5 A \,a^{3} b^{3} d +\frac {3}{2} B \,a^{5} b e +\frac {15}{4} B \,a^{4} b^{2} d \right ) x^{4}+\left (2 A \,a^{5} b e +5 A \,a^{4} b^{2} d +\frac {1}{3} B \,a^{6} e +2 B \,a^{5} b d \right ) x^{3}+\left (\frac {1}{2} A \,a^{6} e +3 A \,a^{5} b d +\frac {1}{2} B \,a^{6} d \right ) x^{2}+A \,a^{6} d x\) | \(279\) |
default | \(\frac {b^{6} B e \,x^{9}}{9}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e +b^{6} B d \right ) x^{8}}{8}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e +\left (b^{6} A +6 a \,b^{5} B \right ) d \right ) x^{7}}{7}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d \right ) x^{6}}{6}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d \right ) x^{5}}{5}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d \right ) x^{4}}{4}+\frac {\left (\left (6 A \,a^{5} b +B \,a^{6}\right ) e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d \right ) x^{3}}{3}+\frac {\left (A \,a^{6} e +\left (6 A \,a^{5} b +B \,a^{6}\right ) d \right ) x^{2}}{2}+A \,a^{6} d x\) | \(293\) |
gosper | \(3 A \,a^{2} b^{4} d \,x^{5}+3 B \,a^{4} b^{2} e \,x^{5}+4 B \,a^{3} b^{3} d \,x^{5}+\frac {1}{2} x^{2} B \,a^{6} d +\frac {1}{2} x^{2} A \,a^{6} e +\frac {1}{3} x^{3} B \,a^{6} e +\frac {1}{7} x^{7} A \,b^{6} d +\frac {1}{8} x^{8} b^{6} B d +\frac {1}{8} x^{8} A \,b^{6} e +A \,a^{6} d x +\frac {1}{9} b^{6} B e \,x^{9}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d +\frac {15}{4} x^{4} A \,a^{4} b^{2} e +5 x^{4} A \,a^{3} b^{3} d +\frac {5}{2} x^{6} A \,a^{2} b^{4} e +x^{6} A a \,b^{5} d +4 A \,a^{3} b^{3} e \,x^{5}+\frac {6}{7} x^{7} A a \,b^{5} e +\frac {10}{3} x^{6} B \,a^{3} b^{3} e +5 x^{3} A \,a^{4} b^{2} d +2 x^{3} B \,a^{5} b d +3 x^{2} A \,a^{5} b d +\frac {3}{4} x^{8} B a \,b^{5} e +\frac {3}{2} x^{4} B \,a^{5} b e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d +2 x^{3} A \,a^{5} b e +\frac {15}{7} x^{7} B \,a^{2} b^{4} e +\frac {6}{7} x^{7} B a \,b^{5} d\) | \(322\) |
risch | \(3 A \,a^{2} b^{4} d \,x^{5}+3 B \,a^{4} b^{2} e \,x^{5}+4 B \,a^{3} b^{3} d \,x^{5}+\frac {1}{2} x^{2} B \,a^{6} d +\frac {1}{2} x^{2} A \,a^{6} e +\frac {1}{3} x^{3} B \,a^{6} e +\frac {1}{7} x^{7} A \,b^{6} d +\frac {1}{8} x^{8} b^{6} B d +\frac {1}{8} x^{8} A \,b^{6} e +A \,a^{6} d x +\frac {1}{9} b^{6} B e \,x^{9}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d +\frac {15}{4} x^{4} A \,a^{4} b^{2} e +5 x^{4} A \,a^{3} b^{3} d +\frac {5}{2} x^{6} A \,a^{2} b^{4} e +x^{6} A a \,b^{5} d +4 A \,a^{3} b^{3} e \,x^{5}+\frac {6}{7} x^{7} A a \,b^{5} e +\frac {10}{3} x^{6} B \,a^{3} b^{3} e +5 x^{3} A \,a^{4} b^{2} d +2 x^{3} B \,a^{5} b d +3 x^{2} A \,a^{5} b d +\frac {3}{4} x^{8} B a \,b^{5} e +\frac {3}{2} x^{4} B \,a^{5} b e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d +2 x^{3} A \,a^{5} b e +\frac {15}{7} x^{7} B \,a^{2} b^{4} e +\frac {6}{7} x^{7} B a \,b^{5} d\) | \(322\) |
parallelrisch | \(3 A \,a^{2} b^{4} d \,x^{5}+3 B \,a^{4} b^{2} e \,x^{5}+4 B \,a^{3} b^{3} d \,x^{5}+\frac {1}{2} x^{2} B \,a^{6} d +\frac {1}{2} x^{2} A \,a^{6} e +\frac {1}{3} x^{3} B \,a^{6} e +\frac {1}{7} x^{7} A \,b^{6} d +\frac {1}{8} x^{8} b^{6} B d +\frac {1}{8} x^{8} A \,b^{6} e +A \,a^{6} d x +\frac {1}{9} b^{6} B e \,x^{9}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d +\frac {15}{4} x^{4} A \,a^{4} b^{2} e +5 x^{4} A \,a^{3} b^{3} d +\frac {5}{2} x^{6} A \,a^{2} b^{4} e +x^{6} A a \,b^{5} d +4 A \,a^{3} b^{3} e \,x^{5}+\frac {6}{7} x^{7} A a \,b^{5} e +\frac {10}{3} x^{6} B \,a^{3} b^{3} e +5 x^{3} A \,a^{4} b^{2} d +2 x^{3} B \,a^{5} b d +3 x^{2} A \,a^{5} b d +\frac {3}{4} x^{8} B a \,b^{5} e +\frac {3}{2} x^{4} B \,a^{5} b e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d +2 x^{3} A \,a^{5} b e +\frac {15}{7} x^{7} B \,a^{2} b^{4} e +\frac {6}{7} x^{7} B a \,b^{5} d\) | \(322\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (69) = 138\).
Time = 0.23 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.96 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {1}{9} \, B b^{6} e x^{9} + A a^{6} d x + \frac {1}{8} \, {\left (B b^{6} d + {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e\right )} x^{6} + {\left ({\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d + {\left (B a^{6} + 6 \, A a^{5} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{6} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (71) = 142\).
Time = 0.04 (sec) , antiderivative size = 333, normalized size of antiderivative = 4.44 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=A a^{6} d x + \frac {B b^{6} e x^{9}}{9} + x^{8} \left (\frac {A b^{6} e}{8} + \frac {3 B a b^{5} e}{4} + \frac {B b^{6} d}{8}\right ) + x^{7} \cdot \left (\frac {6 A a b^{5} e}{7} + \frac {A b^{6} d}{7} + \frac {15 B a^{2} b^{4} e}{7} + \frac {6 B a b^{5} d}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{2} b^{4} e}{2} + A a b^{5} d + \frac {10 B a^{3} b^{3} e}{3} + \frac {5 B a^{2} b^{4} d}{2}\right ) + x^{5} \cdot \left (4 A a^{3} b^{3} e + 3 A a^{2} b^{4} d + 3 B a^{4} b^{2} e + 4 B a^{3} b^{3} d\right ) + x^{4} \cdot \left (\frac {15 A a^{4} b^{2} e}{4} + 5 A a^{3} b^{3} d + \frac {3 B a^{5} b e}{2} + \frac {15 B a^{4} b^{2} d}{4}\right ) + x^{3} \cdot \left (2 A a^{5} b e + 5 A a^{4} b^{2} d + \frac {B a^{6} e}{3} + 2 B a^{5} b d\right ) + x^{2} \left (\frac {A a^{6} e}{2} + 3 A a^{5} b d + \frac {B a^{6} d}{2}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (69) = 138\).
Time = 0.20 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.96 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {1}{9} \, B b^{6} e x^{9} + A a^{6} d x + \frac {1}{8} \, {\left (B b^{6} d + {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e\right )} x^{6} + {\left ({\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d + {\left (B a^{6} + 6 \, A a^{5} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{6} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (69) = 138\).
Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.28 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=\frac {1}{9} \, B b^{6} e x^{9} + \frac {1}{8} \, B b^{6} d x^{8} + \frac {3}{4} \, B a b^{5} e x^{8} + \frac {1}{8} \, A b^{6} e x^{8} + \frac {6}{7} \, B a b^{5} d x^{7} + \frac {1}{7} \, A b^{6} d x^{7} + \frac {15}{7} \, B a^{2} b^{4} e x^{7} + \frac {6}{7} \, A a b^{5} e x^{7} + \frac {5}{2} \, B a^{2} b^{4} d x^{6} + A a b^{5} d x^{6} + \frac {10}{3} \, B a^{3} b^{3} e x^{6} + \frac {5}{2} \, A a^{2} b^{4} e x^{6} + 4 \, B a^{3} b^{3} d x^{5} + 3 \, A a^{2} b^{4} d x^{5} + 3 \, B a^{4} b^{2} e x^{5} + 4 \, A a^{3} b^{3} e x^{5} + \frac {15}{4} \, B a^{4} b^{2} d x^{4} + 5 \, A a^{3} b^{3} d x^{4} + \frac {3}{2} \, B a^{5} b e x^{4} + \frac {15}{4} \, A a^{4} b^{2} e x^{4} + 2 \, B a^{5} b d x^{3} + 5 \, A a^{4} b^{2} d x^{3} + \frac {1}{3} \, B a^{6} e x^{3} + 2 \, A a^{5} b e x^{3} + \frac {1}{2} \, B a^{6} d x^{2} + 3 \, A a^{5} b d x^{2} + \frac {1}{2} \, A a^{6} e x^{2} + A a^{6} d x \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.43 \[ \int (a+b x)^6 (A+B x) (d+e x) \, dx=x^3\,\left (\frac {B\,a^6\,e}{3}+2\,A\,a^5\,b\,e+2\,B\,a^5\,b\,d+5\,A\,a^4\,b^2\,d\right )+x^7\,\left (\frac {A\,b^6\,d}{7}+\frac {6\,A\,a\,b^5\,e}{7}+\frac {6\,B\,a\,b^5\,d}{7}+\frac {15\,B\,a^2\,b^4\,e}{7}\right )+x^2\,\left (\frac {A\,a^6\,e}{2}+\frac {B\,a^6\,d}{2}+3\,A\,a^5\,b\,d\right )+x^8\,\left (\frac {A\,b^6\,e}{8}+\frac {B\,b^6\,d}{8}+\frac {3\,B\,a\,b^5\,e}{4}\right )+a^2\,b^2\,x^5\,\left (3\,A\,b^2\,d+3\,B\,a^2\,e+4\,A\,a\,b\,e+4\,B\,a\,b\,d\right )+A\,a^6\,d\,x+\frac {B\,b^6\,e\,x^9}{9}+\frac {a^3\,b\,x^4\,\left (20\,A\,b^2\,d+6\,B\,a^2\,e+15\,A\,a\,b\,e+15\,B\,a\,b\,d\right )}{4}+\frac {a\,b^3\,x^6\,\left (6\,A\,b^2\,d+20\,B\,a^2\,e+15\,A\,a\,b\,e+15\,B\,a\,b\,d\right )}{6} \]
[In]
[Out]