Integrand size = 20, antiderivative size = 120 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]
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Time = 0.16 (sec) , antiderivative size = 120, 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} \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=-\frac {b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac {(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^4}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^5}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^6}{e^3}+\frac {b^2 B (d+e x)^7}{e^3}\right ) \, dx \\ & = -\frac {(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(120)=240\).
Time = 0.06 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.36 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=a^2 A d^4 x+\frac {1}{2} a d^3 (2 A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 \left (2 a B d (b d+2 a e)+A \left (b^2 d^2+8 a b d e+6 a^2 e^2\right )\right ) x^3+\frac {1}{4} d \left (2 a^2 e^2 (3 B d+2 A e)+4 a b d e (2 B d+3 A e)+b^2 d^2 (B d+4 A e)\right ) x^4+\frac {1}{5} e \left (a^2 e^2 (4 B d+A e)+4 a b d e (3 B d+2 A e)+2 b^2 d^2 (2 B d+3 A e)\right ) x^5+\frac {1}{6} e^2 \left (a^2 B e^2+2 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x^6+\frac {1}{7} b e^3 (4 b B d+A b e+2 a B e) x^7+\frac {1}{8} b^2 B e^4 x^8 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(112)=224\).
Time = 2.01 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.54
method | result | size |
default | \(\frac {b^{2} B \,e^{4} x^{8}}{8}+\frac {\left (\left (b^{2} A +2 a b B \right ) e^{4}+4 b^{2} B d \,e^{3}\right ) x^{7}}{7}+\frac {\left (\left (2 a b A +a^{2} B \right ) e^{4}+4 \left (b^{2} A +2 a b B \right ) d \,e^{3}+6 b^{2} B \,d^{2} e^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} A \,e^{4}+4 \left (2 a b A +a^{2} B \right ) d \,e^{3}+6 \left (b^{2} A +2 a b B \right ) d^{2} e^{2}+4 b^{2} B \,d^{3} e \right ) x^{5}}{5}+\frac {\left (4 a^{2} A d \,e^{3}+6 \left (2 a b A +a^{2} B \right ) d^{2} e^{2}+4 \left (b^{2} A +2 a b B \right ) d^{3} e +b^{2} B \,d^{4}\right ) x^{4}}{4}+\frac {\left (6 a^{2} A \,d^{2} e^{2}+4 \left (2 a b A +a^{2} B \right ) d^{3} e +\left (b^{2} A +2 a b B \right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{2} A \,d^{3} e +\left (2 a b A +a^{2} B \right ) d^{4}\right ) x^{2}}{2}+a^{2} A \,d^{4} x\) | \(305\) |
norman | \(\frac {b^{2} B \,e^{4} x^{8}}{8}+\left (\frac {1}{7} A \,b^{2} e^{4}+\frac {2}{7} B a b \,e^{4}+\frac {4}{7} b^{2} B d \,e^{3}\right ) x^{7}+\left (\frac {1}{3} A a b \,e^{4}+\frac {2}{3} A \,b^{2} d \,e^{3}+\frac {1}{6} B \,a^{2} e^{4}+\frac {4}{3} B a b d \,e^{3}+b^{2} B \,d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} a^{2} A \,e^{4}+\frac {8}{5} A a b d \,e^{3}+\frac {6}{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} B \,a^{2} d \,e^{3}+\frac {12}{5} B a b \,d^{2} e^{2}+\frac {4}{5} b^{2} B \,d^{3} e \right ) x^{5}+\left (a^{2} A d \,e^{3}+3 A a b \,d^{2} e^{2}+A \,b^{2} d^{3} e +\frac {3}{2} B \,a^{2} d^{2} e^{2}+2 B a b \,d^{3} e +\frac {1}{4} b^{2} B \,d^{4}\right ) x^{4}+\left (2 a^{2} A \,d^{2} e^{2}+\frac {8}{3} A a b \,d^{3} e +\frac {1}{3} A \,b^{2} d^{4}+\frac {4}{3} B \,a^{2} d^{3} e +\frac {2}{3} B a b \,d^{4}\right ) x^{3}+\left (2 a^{2} A \,d^{3} e +A a b \,d^{4}+\frac {1}{2} B \,a^{2} d^{4}\right ) x^{2}+a^{2} A \,d^{4} x\) | \(321\) |
gosper | \(\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {1}{6} x^{6} B \,a^{2} e^{4}+\frac {1}{5} x^{5} a^{2} A \,e^{4}+\frac {1}{4} x^{4} b^{2} B \,d^{4}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{8} b^{2} B \,e^{4} x^{8}+a^{2} A \,d^{4} x +\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+\frac {4}{5} x^{5} b^{2} B \,d^{3} e +x^{4} a^{2} A d \,e^{3}+x^{4} A \,b^{2} d^{3} e +\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {4}{7} x^{7} b^{2} B d \,e^{3}+\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} b^{2} B \,d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} B a b \,e^{4}+2 x^{3} a^{2} A \,d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+2 x^{2} a^{2} A \,d^{3} e +x^{2} A a b \,d^{4}\) | \(375\) |
risch | \(\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {1}{6} x^{6} B \,a^{2} e^{4}+\frac {1}{5} x^{5} a^{2} A \,e^{4}+\frac {1}{4} x^{4} b^{2} B \,d^{4}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{8} b^{2} B \,e^{4} x^{8}+a^{2} A \,d^{4} x +\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+\frac {4}{5} x^{5} b^{2} B \,d^{3} e +x^{4} a^{2} A d \,e^{3}+x^{4} A \,b^{2} d^{3} e +\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {4}{7} x^{7} b^{2} B d \,e^{3}+\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} b^{2} B \,d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} B a b \,e^{4}+2 x^{3} a^{2} A \,d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+2 x^{2} a^{2} A \,d^{3} e +x^{2} A a b \,d^{4}\) | \(375\) |
parallelrisch | \(\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {1}{6} x^{6} B \,a^{2} e^{4}+\frac {1}{5} x^{5} a^{2} A \,e^{4}+\frac {1}{4} x^{4} b^{2} B \,d^{4}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{8} b^{2} B \,e^{4} x^{8}+a^{2} A \,d^{4} x +\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+\frac {4}{5} x^{5} b^{2} B \,d^{3} e +x^{4} a^{2} A d \,e^{3}+x^{4} A \,b^{2} d^{3} e +\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {4}{7} x^{7} b^{2} B d \,e^{3}+\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} b^{2} B \,d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} B a b \,e^{4}+2 x^{3} a^{2} A \,d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+2 x^{2} a^{2} A \,d^{3} e +x^{2} A a b \,d^{4}\) | \(375\) |
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (112) = 224\).
Time = 0.21 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (114) = 228\).
Time = 0.04 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.20 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=A a^{2} d^{4} x + \frac {B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{2} e^{4}}{7} + \frac {2 B a b e^{4}}{7} + \frac {4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {2 A b^{2} d e^{3}}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (112) = 224\).
Time = 0.20 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (112) = 224\).
Time = 0.29 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.12 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + \frac {4}{7} \, B b^{2} d e^{3} x^{7} + \frac {2}{7} \, B a b e^{4} x^{7} + \frac {1}{7} \, A b^{2} e^{4} x^{7} + B b^{2} d^{2} e^{2} x^{6} + \frac {4}{3} \, B a b d e^{3} x^{6} + \frac {2}{3} \, A b^{2} d e^{3} x^{6} + \frac {1}{6} \, B a^{2} e^{4} x^{6} + \frac {1}{3} \, A a b e^{4} x^{6} + \frac {4}{5} \, B b^{2} d^{3} e x^{5} + \frac {12}{5} \, B a b d^{2} e^{2} x^{5} + \frac {6}{5} \, A b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{2} d e^{3} x^{5} + \frac {8}{5} \, A a b d e^{3} x^{5} + \frac {1}{5} \, A a^{2} e^{4} x^{5} + \frac {1}{4} \, B b^{2} d^{4} x^{4} + 2 \, B a b d^{3} e x^{4} + A b^{2} d^{3} e x^{4} + \frac {3}{2} \, B a^{2} d^{2} e^{2} x^{4} + 3 \, A a b d^{2} e^{2} x^{4} + A a^{2} d e^{3} x^{4} + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {4}{3} \, B a^{2} d^{3} e x^{3} + \frac {8}{3} \, A a b d^{3} e x^{3} + 2 \, A a^{2} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + 2 \, A a^{2} d^{3} e x^{2} + A a^{2} d^{4} x \]
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Time = 0.14 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.54 \[ \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx=x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}\right )+x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^3\,x^7\,\left (A\,b\,e+2\,B\,a\,e+4\,B\,b\,d\right )}{7}+\frac {B\,b^2\,e^4\,x^8}{8} \]
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