Optimal. Leaf size=215 \[ -\frac {a^7 (A b-a B) (a+b x)^{11}}{11 b^9}+\frac {a^6 (7 A b-8 a B) (a+b x)^{12}}{12 b^9}-\frac {7 a^5 (3 A b-4 a B) (a+b x)^{13}}{13 b^9}+\frac {a^4 (5 A b-8 a B) (a+b x)^{14}}{2 b^9}-\frac {7 a^3 (A b-2 a B) (a+b x)^{15}}{3 b^9}+\frac {7 a^2 (3 A b-8 a B) (a+b x)^{16}}{16 b^9}-\frac {7 a (A b-4 a B) (a+b x)^{17}}{17 b^9}+\frac {(A b-8 a B) (a+b x)^{18}}{18 b^9}+\frac {B (a+b x)^{19}}{19 b^9} \]
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Rubi [A]
time = 0.12, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} -\frac {a^7 (a+b x)^{11} (A b-a B)}{11 b^9}+\frac {a^6 (a+b x)^{12} (7 A b-8 a B)}{12 b^9}-\frac {7 a^5 (a+b x)^{13} (3 A b-4 a B)}{13 b^9}+\frac {a^4 (a+b x)^{14} (5 A b-8 a B)}{2 b^9}-\frac {7 a^3 (a+b x)^{15} (A b-2 a B)}{3 b^9}+\frac {7 a^2 (a+b x)^{16} (3 A b-8 a B)}{16 b^9}+\frac {(a+b x)^{18} (A b-8 a B)}{18 b^9}-\frac {7 a (a+b x)^{17} (A b-4 a B)}{17 b^9}+\frac {B (a+b x)^{19}}{19 b^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^7 (a+b x)^{10} (A+B x) \, dx &=\int \left (\frac {a^7 (-A b+a B) (a+b x)^{10}}{b^8}-\frac {a^6 (-7 A b+8 a B) (a+b x)^{11}}{b^8}+\frac {7 a^5 (-3 A b+4 a B) (a+b x)^{12}}{b^8}-\frac {7 a^4 (-5 A b+8 a B) (a+b x)^{13}}{b^8}+\frac {35 a^3 (-A b+2 a B) (a+b x)^{14}}{b^8}-\frac {7 a^2 (-3 A b+8 a B) (a+b x)^{15}}{b^8}+\frac {7 a (-A b+4 a B) (a+b x)^{16}}{b^8}+\frac {(A b-8 a B) (a+b x)^{17}}{b^8}+\frac {B (a+b x)^{18}}{b^8}\right ) \, dx\\ &=-\frac {a^7 (A b-a B) (a+b x)^{11}}{11 b^9}+\frac {a^6 (7 A b-8 a B) (a+b x)^{12}}{12 b^9}-\frac {7 a^5 (3 A b-4 a B) (a+b x)^{13}}{13 b^9}+\frac {a^4 (5 A b-8 a B) (a+b x)^{14}}{2 b^9}-\frac {7 a^3 (A b-2 a B) (a+b x)^{15}}{3 b^9}+\frac {7 a^2 (3 A b-8 a B) (a+b x)^{16}}{16 b^9}-\frac {7 a (A b-4 a B) (a+b x)^{17}}{17 b^9}+\frac {(A b-8 a B) (a+b x)^{18}}{18 b^9}+\frac {B (a+b x)^{19}}{19 b^9}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 227, normalized size = 1.06 \begin {gather*} \frac {1}{8} a^{10} A x^8+\frac {1}{9} a^9 (10 A b+a B) x^9+\frac {1}{2} a^8 b (9 A b+2 a B) x^{10}+\frac {15}{11} a^7 b^2 (8 A b+3 a B) x^{11}+\frac {5}{2} a^6 b^3 (7 A b+4 a B) x^{12}+\frac {42}{13} a^5 b^4 (6 A b+5 a B) x^{13}+3 a^4 b^5 (5 A b+6 a B) x^{14}+2 a^3 b^6 (4 A b+7 a B) x^{15}+\frac {15}{16} a^2 b^7 (3 A b+8 a B) x^{16}+\frac {5}{17} a b^8 (2 A b+9 a B) x^{17}+\frac {1}{18} b^9 (A b+10 a B) x^{18}+\frac {1}{19} b^{10} B x^{19} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 244, normalized size = 1.13
method | result | size |
norman | \(\frac {a^{10} A \,x^{8}}{8}+\left (\frac {10}{9} a^{9} b A +\frac {1}{9} a^{10} B \right ) x^{9}+\left (\frac {9}{2} a^{8} b^{2} A +a^{9} b B \right ) x^{10}+\left (\frac {120}{11} a^{7} b^{3} A +\frac {45}{11} a^{8} b^{2} B \right ) x^{11}+\left (\frac {35}{2} a^{6} b^{4} A +10 a^{7} b^{3} B \right ) x^{12}+\left (\frac {252}{13} a^{5} b^{5} A +\frac {210}{13} a^{6} b^{4} B \right ) x^{13}+\left (15 a^{4} b^{6} A +18 a^{5} b^{5} B \right ) x^{14}+\left (8 a^{3} b^{7} A +14 a^{4} b^{6} B \right ) x^{15}+\left (\frac {45}{16} a^{2} b^{8} A +\frac {15}{2} a^{3} b^{7} B \right ) x^{16}+\left (\frac {10}{17} a \,b^{9} A +\frac {45}{17} a^{2} b^{8} B \right ) x^{17}+\left (\frac {1}{18} b^{10} A +\frac {5}{9} a \,b^{9} B \right ) x^{18}+\frac {b^{10} B \,x^{19}}{19}\) | \(235\) |
default | \(\frac {b^{10} B \,x^{19}}{19}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{18}}{18}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{17}}{17}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{16}}{16}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{15}}{15}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{14}}{14}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{13}}{13}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{12}}{12}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{11}}{11}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{10}}{10}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{9}}{9}+\frac {a^{10} A \,x^{8}}{8}\) | \(244\) |
gosper | \(\frac {1}{8} a^{10} A \,x^{8}+\frac {10}{9} x^{9} a^{9} b A +\frac {1}{9} x^{9} a^{10} B +\frac {9}{2} x^{10} a^{8} b^{2} A +x^{10} a^{9} b B +\frac {120}{11} x^{11} a^{7} b^{3} A +\frac {45}{11} x^{11} a^{8} b^{2} B +\frac {35}{2} x^{12} a^{6} b^{4} A +10 x^{12} a^{7} b^{3} B +\frac {252}{13} x^{13} a^{5} b^{5} A +\frac {210}{13} x^{13} a^{6} b^{4} B +15 A \,a^{4} b^{6} x^{14}+18 B \,a^{5} b^{5} x^{14}+8 A \,a^{3} b^{7} x^{15}+14 B \,a^{4} b^{6} x^{15}+\frac {45}{16} x^{16} a^{2} b^{8} A +\frac {15}{2} x^{16} a^{3} b^{7} B +\frac {10}{17} x^{17} a \,b^{9} A +\frac {45}{17} x^{17} a^{2} b^{8} B +\frac {1}{18} x^{18} b^{10} A +\frac {5}{9} x^{18} a \,b^{9} B +\frac {1}{19} b^{10} B \,x^{19}\) | \(245\) |
risch | \(\frac {1}{8} a^{10} A \,x^{8}+\frac {10}{9} x^{9} a^{9} b A +\frac {1}{9} x^{9} a^{10} B +\frac {9}{2} x^{10} a^{8} b^{2} A +x^{10} a^{9} b B +\frac {120}{11} x^{11} a^{7} b^{3} A +\frac {45}{11} x^{11} a^{8} b^{2} B +\frac {35}{2} x^{12} a^{6} b^{4} A +10 x^{12} a^{7} b^{3} B +\frac {252}{13} x^{13} a^{5} b^{5} A +\frac {210}{13} x^{13} a^{6} b^{4} B +15 A \,a^{4} b^{6} x^{14}+18 B \,a^{5} b^{5} x^{14}+8 A \,a^{3} b^{7} x^{15}+14 B \,a^{4} b^{6} x^{15}+\frac {45}{16} x^{16} a^{2} b^{8} A +\frac {15}{2} x^{16} a^{3} b^{7} B +\frac {10}{17} x^{17} a \,b^{9} A +\frac {45}{17} x^{17} a^{2} b^{8} B +\frac {1}{18} x^{18} b^{10} A +\frac {5}{9} x^{18} a \,b^{9} B +\frac {1}{19} b^{10} B \,x^{19}\) | \(245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 243, normalized size = 1.13 \begin {gather*} \frac {1}{19} \, B b^{10} x^{19} + \frac {1}{8} \, A a^{10} x^{8} + \frac {1}{18} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{18} + \frac {5}{17} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{17} + \frac {15}{16} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{16} + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{15} + 3 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{14} + \frac {42}{13} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{13} + \frac {5}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{12} + \frac {15}{11} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{10} + \frac {1}{9} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 243, normalized size = 1.13 \begin {gather*} \frac {1}{19} \, B b^{10} x^{19} + \frac {1}{8} \, A a^{10} x^{8} + \frac {1}{18} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{18} + \frac {5}{17} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{17} + \frac {15}{16} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{16} + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{15} + 3 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{14} + \frac {42}{13} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{13} + \frac {5}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{12} + \frac {15}{11} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{10} + \frac {1}{9} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 262, normalized size = 1.22 \begin {gather*} \frac {A a^{10} x^{8}}{8} + \frac {B b^{10} x^{19}}{19} + x^{18} \left (\frac {A b^{10}}{18} + \frac {5 B a b^{9}}{9}\right ) + x^{17} \cdot \left (\frac {10 A a b^{9}}{17} + \frac {45 B a^{2} b^{8}}{17}\right ) + x^{16} \cdot \left (\frac {45 A a^{2} b^{8}}{16} + \frac {15 B a^{3} b^{7}}{2}\right ) + x^{15} \cdot \left (8 A a^{3} b^{7} + 14 B a^{4} b^{6}\right ) + x^{14} \cdot \left (15 A a^{4} b^{6} + 18 B a^{5} b^{5}\right ) + x^{13} \cdot \left (\frac {252 A a^{5} b^{5}}{13} + \frac {210 B a^{6} b^{4}}{13}\right ) + x^{12} \cdot \left (\frac {35 A a^{6} b^{4}}{2} + 10 B a^{7} b^{3}\right ) + x^{11} \cdot \left (\frac {120 A a^{7} b^{3}}{11} + \frac {45 B a^{8} b^{2}}{11}\right ) + x^{10} \cdot \left (\frac {9 A a^{8} b^{2}}{2} + B a^{9} b\right ) + x^{9} \cdot \left (\frac {10 A a^{9} b}{9} + \frac {B a^{10}}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 244, normalized size = 1.13 \begin {gather*} \frac {1}{19} \, B b^{10} x^{19} + \frac {5}{9} \, B a b^{9} x^{18} + \frac {1}{18} \, A b^{10} x^{18} + \frac {45}{17} \, B a^{2} b^{8} x^{17} + \frac {10}{17} \, A a b^{9} x^{17} + \frac {15}{2} \, B a^{3} b^{7} x^{16} + \frac {45}{16} \, A a^{2} b^{8} x^{16} + 14 \, B a^{4} b^{6} x^{15} + 8 \, A a^{3} b^{7} x^{15} + 18 \, B a^{5} b^{5} x^{14} + 15 \, A a^{4} b^{6} x^{14} + \frac {210}{13} \, B a^{6} b^{4} x^{13} + \frac {252}{13} \, A a^{5} b^{5} x^{13} + 10 \, B a^{7} b^{3} x^{12} + \frac {35}{2} \, A a^{6} b^{4} x^{12} + \frac {45}{11} \, B a^{8} b^{2} x^{11} + \frac {120}{11} \, A a^{7} b^{3} x^{11} + B a^{9} b x^{10} + \frac {9}{2} \, A a^{8} b^{2} x^{10} + \frac {1}{9} \, B a^{10} x^{9} + \frac {10}{9} \, A a^{9} b x^{9} + \frac {1}{8} \, A a^{10} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 211, normalized size = 0.98 \begin {gather*} x^9\,\left (\frac {B\,a^{10}}{9}+\frac {10\,A\,b\,a^9}{9}\right )+x^{18}\,\left (\frac {A\,b^{10}}{18}+\frac {5\,B\,a\,b^9}{9}\right )+\frac {A\,a^{10}\,x^8}{8}+\frac {B\,b^{10}\,x^{19}}{19}+\frac {15\,a^7\,b^2\,x^{11}\,\left (8\,A\,b+3\,B\,a\right )}{11}+\frac {5\,a^6\,b^3\,x^{12}\,\left (7\,A\,b+4\,B\,a\right )}{2}+\frac {42\,a^5\,b^4\,x^{13}\,\left (6\,A\,b+5\,B\,a\right )}{13}+3\,a^4\,b^5\,x^{14}\,\left (5\,A\,b+6\,B\,a\right )+2\,a^3\,b^6\,x^{15}\,\left (4\,A\,b+7\,B\,a\right )+\frac {15\,a^2\,b^7\,x^{16}\,\left (3\,A\,b+8\,B\,a\right )}{16}+\frac {a^8\,b\,x^{10}\,\left (9\,A\,b+2\,B\,a\right )}{2}+\frac {5\,a\,b^8\,x^{17}\,\left (2\,A\,b+9\,B\,a\right )}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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