Optimal. Leaf size=163 \[ -\frac {a^5 (A b-a B) (a+b x)^{11}}{11 b^7}+\frac {a^4 (5 A b-6 a B) (a+b x)^{12}}{12 b^7}-\frac {5 a^3 (2 A b-3 a B) (a+b x)^{13}}{13 b^7}+\frac {5 a^2 (A b-2 a B) (a+b x)^{14}}{7 b^7}-\frac {a (A b-3 a B) (a+b x)^{15}}{3 b^7}+\frac {(A b-6 a B) (a+b x)^{16}}{16 b^7}+\frac {B (a+b x)^{17}}{17 b^7} \]
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Rubi [A]
time = 0.09, antiderivative size = 163, 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^5 (a+b x)^{11} (A b-a B)}{11 b^7}+\frac {a^4 (a+b x)^{12} (5 A b-6 a B)}{12 b^7}-\frac {5 a^3 (a+b x)^{13} (2 A b-3 a B)}{13 b^7}+\frac {5 a^2 (a+b x)^{14} (A b-2 a B)}{7 b^7}+\frac {(a+b x)^{16} (A b-6 a B)}{16 b^7}-\frac {a (a+b x)^{15} (A b-3 a B)}{3 b^7}+\frac {B (a+b x)^{17}}{17 b^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^5 (a+b x)^{10} (A+B x) \, dx &=\int \left (\frac {a^5 (-A b+a B) (a+b x)^{10}}{b^6}-\frac {a^4 (-5 A b+6 a B) (a+b x)^{11}}{b^6}+\frac {5 a^3 (-2 A b+3 a B) (a+b x)^{12}}{b^6}-\frac {10 a^2 (-A b+2 a B) (a+b x)^{13}}{b^6}+\frac {5 a (-A b+3 a B) (a+b x)^{14}}{b^6}+\frac {(A b-6 a B) (a+b x)^{15}}{b^6}+\frac {B (a+b x)^{16}}{b^6}\right ) \, dx\\ &=-\frac {a^5 (A b-a B) (a+b x)^{11}}{11 b^7}+\frac {a^4 (5 A b-6 a B) (a+b x)^{12}}{12 b^7}-\frac {5 a^3 (2 A b-3 a B) (a+b x)^{13}}{13 b^7}+\frac {5 a^2 (A b-2 a B) (a+b x)^{14}}{7 b^7}-\frac {a (A b-3 a B) (a+b x)^{15}}{3 b^7}+\frac {(A b-6 a B) (a+b x)^{16}}{16 b^7}+\frac {B (a+b x)^{17}}{17 b^7}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 229, normalized size = 1.40 \begin {gather*} \frac {1}{6} a^{10} A x^6+\frac {1}{7} a^9 (10 A b+a B) x^7+\frac {5}{8} a^8 b (9 A b+2 a B) x^8+\frac {5}{3} a^7 b^2 (8 A b+3 a B) x^9+3 a^6 b^3 (7 A b+4 a B) x^{10}+\frac {42}{11} a^5 b^4 (6 A b+5 a B) x^{11}+\frac {7}{2} a^4 b^5 (5 A b+6 a B) x^{12}+\frac {30}{13} a^3 b^6 (4 A b+7 a B) x^{13}+\frac {15}{14} a^2 b^7 (3 A b+8 a B) x^{14}+\frac {1}{3} a b^8 (2 A b+9 a B) x^{15}+\frac {1}{16} b^9 (A b+10 a B) x^{16}+\frac {1}{17} b^{10} B x^{17} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 244, normalized size = 1.50
method | result | size |
norman | \(\frac {a^{10} A \,x^{6}}{6}+\left (\frac {10}{7} a^{9} b A +\frac {1}{7} a^{10} B \right ) x^{7}+\left (\frac {45}{8} a^{8} b^{2} A +\frac {5}{4} a^{9} b B \right ) x^{8}+\left (\frac {40}{3} a^{7} b^{3} A +5 a^{8} b^{2} B \right ) x^{9}+\left (21 a^{6} b^{4} A +12 a^{7} b^{3} B \right ) x^{10}+\left (\frac {252}{11} a^{5} b^{5} A +\frac {210}{11} a^{6} b^{4} B \right ) x^{11}+\left (\frac {35}{2} a^{4} b^{6} A +21 a^{5} b^{5} B \right ) x^{12}+\left (\frac {120}{13} a^{3} b^{7} A +\frac {210}{13} a^{4} b^{6} B \right ) x^{13}+\left (\frac {45}{14} a^{2} b^{8} A +\frac {60}{7} a^{3} b^{7} B \right ) x^{14}+\left (\frac {2}{3} a \,b^{9} A +3 a^{2} b^{8} B \right ) x^{15}+\left (\frac {1}{16} b^{10} A +\frac {5}{8} a \,b^{9} B \right ) x^{16}+\frac {b^{10} B \,x^{17}}{17}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{17}}{17}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{16}}{16}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{15}}{15}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{14}}{14}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{13}}{13}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{12}}{12}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{11}}{11}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{10}}{10}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{9}}{9}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{8}}{8}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{7}}{7}+\frac {a^{10} A \,x^{6}}{6}\) | \(244\) |
gosper | \(\frac {1}{6} a^{10} A \,x^{6}+\frac {10}{7} x^{7} a^{9} b A +\frac {1}{7} x^{7} a^{10} B +\frac {45}{8} x^{8} a^{8} b^{2} A +\frac {5}{4} x^{8} a^{9} b B +\frac {40}{3} x^{9} a^{7} b^{3} A +5 x^{9} a^{8} b^{2} B +21 A \,a^{6} b^{4} x^{10}+12 B \,a^{7} b^{3} x^{10}+\frac {252}{11} x^{11} a^{5} b^{5} A +\frac {210}{11} x^{11} a^{6} b^{4} B +\frac {35}{2} x^{12} a^{4} b^{6} A +21 x^{12} a^{5} b^{5} B +\frac {120}{13} x^{13} a^{3} b^{7} A +\frac {210}{13} x^{13} a^{4} b^{6} B +\frac {45}{14} x^{14} a^{2} b^{8} A +\frac {60}{7} x^{14} a^{3} b^{7} B +\frac {2}{3} x^{15} a \,b^{9} A +3 x^{15} a^{2} b^{8} B +\frac {1}{16} x^{16} b^{10} A +\frac {5}{8} x^{16} a \,b^{9} B +\frac {1}{17} b^{10} B \,x^{17}\) | \(246\) |
risch | \(\frac {1}{6} a^{10} A \,x^{6}+\frac {10}{7} x^{7} a^{9} b A +\frac {1}{7} x^{7} a^{10} B +\frac {45}{8} x^{8} a^{8} b^{2} A +\frac {5}{4} x^{8} a^{9} b B +\frac {40}{3} x^{9} a^{7} b^{3} A +5 x^{9} a^{8} b^{2} B +21 A \,a^{6} b^{4} x^{10}+12 B \,a^{7} b^{3} x^{10}+\frac {252}{11} x^{11} a^{5} b^{5} A +\frac {210}{11} x^{11} a^{6} b^{4} B +\frac {35}{2} x^{12} a^{4} b^{6} A +21 x^{12} a^{5} b^{5} B +\frac {120}{13} x^{13} a^{3} b^{7} A +\frac {210}{13} x^{13} a^{4} b^{6} B +\frac {45}{14} x^{14} a^{2} b^{8} A +\frac {60}{7} x^{14} a^{3} b^{7} B +\frac {2}{3} x^{15} a \,b^{9} A +3 x^{15} a^{2} b^{8} B +\frac {1}{16} x^{16} b^{10} A +\frac {5}{8} x^{16} a \,b^{9} B +\frac {1}{17} b^{10} B \,x^{17}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 243, normalized size = 1.49 \begin {gather*} \frac {1}{17} \, B b^{10} x^{17} + \frac {1}{6} \, A a^{10} x^{6} + \frac {1}{16} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{16} + \frac {1}{3} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{15} + \frac {15}{14} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{14} + \frac {30}{13} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{13} + \frac {7}{2} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{12} + \frac {42}{11} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{11} + 3 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{10} + \frac {5}{3} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.78, size = 243, normalized size = 1.49 \begin {gather*} \frac {1}{17} \, B b^{10} x^{17} + \frac {1}{6} \, A a^{10} x^{6} + \frac {1}{16} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{16} + \frac {1}{3} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{15} + \frac {15}{14} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{14} + \frac {30}{13} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{13} + \frac {7}{2} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{12} + \frac {42}{11} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{11} + 3 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{10} + \frac {5}{3} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 265, normalized size = 1.63 \begin {gather*} \frac {A a^{10} x^{6}}{6} + \frac {B b^{10} x^{17}}{17} + x^{16} \left (\frac {A b^{10}}{16} + \frac {5 B a b^{9}}{8}\right ) + x^{15} \cdot \left (\frac {2 A a b^{9}}{3} + 3 B a^{2} b^{8}\right ) + x^{14} \cdot \left (\frac {45 A a^{2} b^{8}}{14} + \frac {60 B a^{3} b^{7}}{7}\right ) + x^{13} \cdot \left (\frac {120 A a^{3} b^{7}}{13} + \frac {210 B a^{4} b^{6}}{13}\right ) + x^{12} \cdot \left (\frac {35 A a^{4} b^{6}}{2} + 21 B a^{5} b^{5}\right ) + x^{11} \cdot \left (\frac {252 A a^{5} b^{5}}{11} + \frac {210 B a^{6} b^{4}}{11}\right ) + x^{10} \cdot \left (21 A a^{6} b^{4} + 12 B a^{7} b^{3}\right ) + x^{9} \cdot \left (\frac {40 A a^{7} b^{3}}{3} + 5 B a^{8} b^{2}\right ) + x^{8} \cdot \left (\frac {45 A a^{8} b^{2}}{8} + \frac {5 B a^{9} b}{4}\right ) + x^{7} \cdot \left (\frac {10 A a^{9} b}{7} + \frac {B a^{10}}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 245, normalized size = 1.50 \begin {gather*} \frac {1}{17} \, B b^{10} x^{17} + \frac {5}{8} \, B a b^{9} x^{16} + \frac {1}{16} \, A b^{10} x^{16} + 3 \, B a^{2} b^{8} x^{15} + \frac {2}{3} \, A a b^{9} x^{15} + \frac {60}{7} \, B a^{3} b^{7} x^{14} + \frac {45}{14} \, A a^{2} b^{8} x^{14} + \frac {210}{13} \, B a^{4} b^{6} x^{13} + \frac {120}{13} \, A a^{3} b^{7} x^{13} + 21 \, B a^{5} b^{5} x^{12} + \frac {35}{2} \, A a^{4} b^{6} x^{12} + \frac {210}{11} \, B a^{6} b^{4} x^{11} + \frac {252}{11} \, A a^{5} b^{5} x^{11} + 12 \, B a^{7} b^{3} x^{10} + 21 \, A a^{6} b^{4} x^{10} + 5 \, B a^{8} b^{2} x^{9} + \frac {40}{3} \, A a^{7} b^{3} x^{9} + \frac {5}{4} \, B a^{9} b x^{8} + \frac {45}{8} \, A a^{8} b^{2} x^{8} + \frac {1}{7} \, B a^{10} x^{7} + \frac {10}{7} \, A a^{9} b x^{7} + \frac {1}{6} \, A a^{10} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 211, normalized size = 1.29 \begin {gather*} x^7\,\left (\frac {B\,a^{10}}{7}+\frac {10\,A\,b\,a^9}{7}\right )+x^{16}\,\left (\frac {A\,b^{10}}{16}+\frac {5\,B\,a\,b^9}{8}\right )+\frac {A\,a^{10}\,x^6}{6}+\frac {B\,b^{10}\,x^{17}}{17}+\frac {5\,a^7\,b^2\,x^9\,\left (8\,A\,b+3\,B\,a\right )}{3}+3\,a^6\,b^3\,x^{10}\,\left (7\,A\,b+4\,B\,a\right )+\frac {42\,a^5\,b^4\,x^{11}\,\left (6\,A\,b+5\,B\,a\right )}{11}+\frac {7\,a^4\,b^5\,x^{12}\,\left (5\,A\,b+6\,B\,a\right )}{2}+\frac {30\,a^3\,b^6\,x^{13}\,\left (4\,A\,b+7\,B\,a\right )}{13}+\frac {15\,a^2\,b^7\,x^{14}\,\left (3\,A\,b+8\,B\,a\right )}{14}+\frac {5\,a^8\,b\,x^8\,\left (9\,A\,b+2\,B\,a\right )}{8}+\frac {a\,b^8\,x^{15}\,\left (2\,A\,b+9\,B\,a\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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