Optimal. Leaf size=118 \[ \frac {e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac {(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4} \]
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Rubi [A] time = 0.33, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac {(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^2 (a+b x)^6}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^7}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^8}{b^3}+\frac {B e^2 (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^7}{7 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^8}{8 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^9}{9 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 386, normalized size = 3.27 \[ \frac {x \left (210 a^6 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+252 a^5 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+630 a^4 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+120 a^3 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+45 a^2 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+30 a b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )\right )}{2520} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 552, normalized size = 4.68 \[ \frac {1}{10} x^{10} e^{2} b^{6} B + \frac {2}{9} x^{9} e d b^{6} B + \frac {2}{3} x^{9} e^{2} b^{5} a B + \frac {1}{9} x^{9} e^{2} b^{6} A + \frac {1}{8} x^{8} d^{2} b^{6} B + \frac {3}{2} x^{8} e d b^{5} a B + \frac {15}{8} x^{8} e^{2} b^{4} a^{2} B + \frac {1}{4} x^{8} e d b^{6} A + \frac {3}{4} x^{8} e^{2} b^{5} a A + \frac {6}{7} x^{7} d^{2} b^{5} a B + \frac {30}{7} x^{7} e d b^{4} a^{2} B + \frac {20}{7} x^{7} e^{2} b^{3} a^{3} B + \frac {1}{7} x^{7} d^{2} b^{6} A + \frac {12}{7} x^{7} e d b^{5} a A + \frac {15}{7} x^{7} e^{2} b^{4} a^{2} A + \frac {5}{2} x^{6} d^{2} b^{4} a^{2} B + \frac {20}{3} x^{6} e d b^{3} a^{3} B + \frac {5}{2} x^{6} e^{2} b^{2} a^{4} B + x^{6} d^{2} b^{5} a A + 5 x^{6} e d b^{4} a^{2} A + \frac {10}{3} x^{6} e^{2} b^{3} a^{3} A + 4 x^{5} d^{2} b^{3} a^{3} B + 6 x^{5} e d b^{2} a^{4} B + \frac {6}{5} x^{5} e^{2} b a^{5} B + 3 x^{5} d^{2} b^{4} a^{2} A + 8 x^{5} e d b^{3} a^{3} A + 3 x^{5} e^{2} b^{2} a^{4} A + \frac {15}{4} x^{4} d^{2} b^{2} a^{4} B + 3 x^{4} e d b a^{5} B + \frac {1}{4} x^{4} e^{2} a^{6} B + 5 x^{4} d^{2} b^{3} a^{3} A + \frac {15}{2} x^{4} e d b^{2} a^{4} A + \frac {3}{2} x^{4} e^{2} b a^{5} A + 2 x^{3} d^{2} b a^{5} B + \frac {2}{3} x^{3} e d a^{6} B + 5 x^{3} d^{2} b^{2} a^{4} A + 4 x^{3} e d b a^{5} A + \frac {1}{3} x^{3} e^{2} a^{6} A + \frac {1}{2} x^{2} d^{2} a^{6} B + 3 x^{2} d^{2} b a^{5} A + x^{2} e d a^{6} A + x d^{2} a^{6} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 552, normalized size = 4.68 \[ \frac {1}{10} \, B b^{6} x^{10} e^{2} + \frac {2}{9} \, B b^{6} d x^{9} e + \frac {1}{8} \, B b^{6} d^{2} x^{8} + \frac {2}{3} \, B a b^{5} x^{9} e^{2} + \frac {1}{9} \, A b^{6} x^{9} e^{2} + \frac {3}{2} \, B a b^{5} d x^{8} e + \frac {1}{4} \, A b^{6} d x^{8} e + \frac {6}{7} \, B a b^{5} d^{2} x^{7} + \frac {1}{7} \, A b^{6} d^{2} x^{7} + \frac {15}{8} \, B a^{2} b^{4} x^{8} e^{2} + \frac {3}{4} \, A a b^{5} x^{8} e^{2} + \frac {30}{7} \, B a^{2} b^{4} d x^{7} e + \frac {12}{7} \, A a b^{5} d x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d^{2} x^{6} + A a b^{5} d^{2} x^{6} + \frac {20}{7} \, B a^{3} b^{3} x^{7} e^{2} + \frac {15}{7} \, A a^{2} b^{4} x^{7} e^{2} + \frac {20}{3} \, B a^{3} b^{3} d x^{6} e + 5 \, A a^{2} b^{4} d x^{6} e + 4 \, B a^{3} b^{3} d^{2} x^{5} + 3 \, A a^{2} b^{4} d^{2} x^{5} + \frac {5}{2} \, B a^{4} b^{2} x^{6} e^{2} + \frac {10}{3} \, A a^{3} b^{3} x^{6} e^{2} + 6 \, B a^{4} b^{2} d x^{5} e + 8 \, A a^{3} b^{3} d x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d^{2} x^{4} + 5 \, A a^{3} b^{3} d^{2} x^{4} + \frac {6}{5} \, B a^{5} b x^{5} e^{2} + 3 \, A a^{4} b^{2} x^{5} e^{2} + 3 \, B a^{5} b d x^{4} e + \frac {15}{2} \, A a^{4} b^{2} d x^{4} e + 2 \, B a^{5} b d^{2} x^{3} + 5 \, A a^{4} b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{6} x^{4} e^{2} + \frac {3}{2} \, A a^{5} b x^{4} e^{2} + \frac {2}{3} \, B a^{6} d x^{3} e + 4 \, A a^{5} b d x^{3} e + \frac {1}{2} \, B a^{6} d^{2} x^{2} + 3 \, A a^{5} b d^{2} x^{2} + \frac {1}{3} \, A a^{6} x^{3} e^{2} + A a^{6} d x^{2} e + A a^{6} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 469, normalized size = 3.97 \[ \frac {B \,b^{6} e^{2} x^{10}}{10}+A \,a^{6} d^{2} x +\frac {\left (2 B \,b^{6} d e +\left (b^{6} A +6 a \,b^{5} B \right ) e^{2}\right ) x^{9}}{9}+\frac {\left (B \,b^{6} d^{2}+2 \left (b^{6} A +6 a \,b^{5} B \right ) d e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{2}\right ) x^{8}}{8}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) d^{2}+2 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{2}\right ) x^{7}}{7}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2}+2 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{2}\right ) x^{6}}{6}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2}+2 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2}+2 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d e +\left (6 a^{5} b A +a^{6} B \right ) e^{2}\right ) x^{4}}{4}+\frac {\left (A \,a^{6} e^{2}+\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2}+2 \left (6 a^{5} b A +a^{6} B \right ) d e \right ) x^{3}}{3}+\frac {\left (2 A \,a^{6} d e +\left (6 a^{5} b A +a^{6} B \right ) d^{2}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 476, normalized size = 4.03 \[ \frac {1}{10} \, B b^{6} e^{2} x^{10} + A a^{6} d^{2} x + \frac {1}{9} \, {\left (2 \, B b^{6} d e + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{2} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{6} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{6} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 457, normalized size = 3.87 \[ x^4\,\left (\frac {B\,a^6\,e^2}{4}+3\,B\,a^5\,b\,d\,e+\frac {3\,A\,a^5\,b\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^2}{4}+\frac {15\,A\,a^4\,b^2\,d\,e}{2}+5\,A\,a^3\,b^3\,d^2\right )+x^7\,\left (\frac {20\,B\,a^3\,b^3\,e^2}{7}+\frac {30\,B\,a^2\,b^4\,d\,e}{7}+\frac {15\,A\,a^2\,b^4\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^2}{7}+\frac {12\,A\,a\,b^5\,d\,e}{7}+\frac {A\,b^6\,d^2}{7}\right )+x^5\,\left (\frac {6\,B\,a^5\,b\,e^2}{5}+6\,B\,a^4\,b^2\,d\,e+3\,A\,a^4\,b^2\,e^2+4\,B\,a^3\,b^3\,d^2+8\,A\,a^3\,b^3\,d\,e+3\,A\,a^2\,b^4\,d^2\right )+x^6\,\left (\frac {5\,B\,a^4\,b^2\,e^2}{2}+\frac {20\,B\,a^3\,b^3\,d\,e}{3}+\frac {10\,A\,a^3\,b^3\,e^2}{3}+\frac {5\,B\,a^2\,b^4\,d^2}{2}+5\,A\,a^2\,b^4\,d\,e+A\,a\,b^5\,d^2\right )+x^3\,\left (\frac {2\,B\,a^6\,d\,e}{3}+\frac {A\,a^6\,e^2}{3}+2\,B\,a^5\,b\,d^2+4\,A\,a^5\,b\,d\,e+5\,A\,a^4\,b^2\,d^2\right )+x^8\,\left (\frac {15\,B\,a^2\,b^4\,e^2}{8}+\frac {3\,B\,a\,b^5\,d\,e}{2}+\frac {3\,A\,a\,b^5\,e^2}{4}+\frac {B\,b^6\,d^2}{8}+\frac {A\,b^6\,d\,e}{4}\right )+A\,a^6\,d^2\,x+\frac {a^5\,d\,x^2\,\left (2\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e\,x^9\,\left (A\,b\,e+6\,B\,a\,e+2\,B\,b\,d\right )}{9}+\frac {B\,b^6\,e^2\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 568, normalized size = 4.81 \[ A a^{6} d^{2} x + \frac {B b^{6} e^{2} x^{10}}{10} + x^{9} \left (\frac {A b^{6} e^{2}}{9} + \frac {2 B a b^{5} e^{2}}{3} + \frac {2 B b^{6} d e}{9}\right ) + x^{8} \left (\frac {3 A a b^{5} e^{2}}{4} + \frac {A b^{6} d e}{4} + \frac {15 B a^{2} b^{4} e^{2}}{8} + \frac {3 B a b^{5} d e}{2} + \frac {B b^{6} d^{2}}{8}\right ) + x^{7} \left (\frac {15 A a^{2} b^{4} e^{2}}{7} + \frac {12 A a b^{5} d e}{7} + \frac {A b^{6} d^{2}}{7} + \frac {20 B a^{3} b^{3} e^{2}}{7} + \frac {30 B a^{2} b^{4} d e}{7} + \frac {6 B a b^{5} d^{2}}{7}\right ) + x^{6} \left (\frac {10 A a^{3} b^{3} e^{2}}{3} + 5 A a^{2} b^{4} d e + A a b^{5} d^{2} + \frac {5 B a^{4} b^{2} e^{2}}{2} + \frac {20 B a^{3} b^{3} d e}{3} + \frac {5 B a^{2} b^{4} d^{2}}{2}\right ) + x^{5} \left (3 A a^{4} b^{2} e^{2} + 8 A a^{3} b^{3} d e + 3 A a^{2} b^{4} d^{2} + \frac {6 B a^{5} b e^{2}}{5} + 6 B a^{4} b^{2} d e + 4 B a^{3} b^{3} d^{2}\right ) + x^{4} \left (\frac {3 A a^{5} b e^{2}}{2} + \frac {15 A a^{4} b^{2} d e}{2} + 5 A a^{3} b^{3} d^{2} + \frac {B a^{6} e^{2}}{4} + 3 B a^{5} b d e + \frac {15 B a^{4} b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{6} e^{2}}{3} + 4 A a^{5} b d e + 5 A a^{4} b^{2} d^{2} + \frac {2 B a^{6} d e}{3} + 2 B a^{5} b d^{2}\right ) + x^{2} \left (A a^{6} d e + 3 A a^{5} b d^{2} + \frac {B a^{6} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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