Integrand size = 15, antiderivative size = 242 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=-\frac {a^{11} \left (a+b \sqrt {x}\right )^{16}}{8 b^{12}}+\frac {22 a^{10} \left (a+b \sqrt {x}\right )^{17}}{17 b^{12}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{18}}{9 b^{12}}+\frac {330 a^8 \left (a+b \sqrt {x}\right )^{19}}{19 b^{12}}-\frac {33 a^7 \left (a+b \sqrt {x}\right )^{20}}{b^{12}}+\frac {44 a^6 \left (a+b \sqrt {x}\right )^{21}}{b^{12}}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{22}}{b^{12}}+\frac {660 a^4 \left (a+b \sqrt {x}\right )^{23}}{23 b^{12}}-\frac {55 a^3 \left (a+b \sqrt {x}\right )^{24}}{4 b^{12}}+\frac {22 a^2 \left (a+b \sqrt {x}\right )^{25}}{5 b^{12}}-\frac {11 a \left (a+b \sqrt {x}\right )^{26}}{13 b^{12}}+\frac {2 \left (a+b \sqrt {x}\right )^{27}}{27 b^{12}} \]
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Time = 0.11 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=-\frac {a^{11} \left (a+b \sqrt {x}\right )^{16}}{8 b^{12}}+\frac {22 a^{10} \left (a+b \sqrt {x}\right )^{17}}{17 b^{12}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{18}}{9 b^{12}}+\frac {330 a^8 \left (a+b \sqrt {x}\right )^{19}}{19 b^{12}}-\frac {33 a^7 \left (a+b \sqrt {x}\right )^{20}}{b^{12}}+\frac {44 a^6 \left (a+b \sqrt {x}\right )^{21}}{b^{12}}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{22}}{b^{12}}+\frac {660 a^4 \left (a+b \sqrt {x}\right )^{23}}{23 b^{12}}-\frac {55 a^3 \left (a+b \sqrt {x}\right )^{24}}{4 b^{12}}+\frac {22 a^2 \left (a+b \sqrt {x}\right )^{25}}{5 b^{12}}+\frac {2 \left (a+b \sqrt {x}\right )^{27}}{27 b^{12}}-\frac {11 a \left (a+b \sqrt {x}\right )^{26}}{13 b^{12}} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^{11} (a+b x)^{15} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a^{11} (a+b x)^{15}}{b^{11}}+\frac {11 a^{10} (a+b x)^{16}}{b^{11}}-\frac {55 a^9 (a+b x)^{17}}{b^{11}}+\frac {165 a^8 (a+b x)^{18}}{b^{11}}-\frac {330 a^7 (a+b x)^{19}}{b^{11}}+\frac {462 a^6 (a+b x)^{20}}{b^{11}}-\frac {462 a^5 (a+b x)^{21}}{b^{11}}+\frac {330 a^4 (a+b x)^{22}}{b^{11}}-\frac {165 a^3 (a+b x)^{23}}{b^{11}}+\frac {55 a^2 (a+b x)^{24}}{b^{11}}-\frac {11 a (a+b x)^{25}}{b^{11}}+\frac {(a+b x)^{26}}{b^{11}}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{11} \left (a+b \sqrt {x}\right )^{16}}{8 b^{12}}+\frac {22 a^{10} \left (a+b \sqrt {x}\right )^{17}}{17 b^{12}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{18}}{9 b^{12}}+\frac {330 a^8 \left (a+b \sqrt {x}\right )^{19}}{19 b^{12}}-\frac {33 a^7 \left (a+b \sqrt {x}\right )^{20}}{b^{12}}+\frac {44 a^6 \left (a+b \sqrt {x}\right )^{21}}{b^{12}}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{22}}{b^{12}}+\frac {660 a^4 \left (a+b \sqrt {x}\right )^{23}}{23 b^{12}}-\frac {55 a^3 \left (a+b \sqrt {x}\right )^{24}}{4 b^{12}}+\frac {22 a^2 \left (a+b \sqrt {x}\right )^{25}}{5 b^{12}}-\frac {11 a \left (a+b \sqrt {x}\right )^{26}}{13 b^{12}}+\frac {2 \left (a+b \sqrt {x}\right )^{27}}{27 b^{12}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {17383860 a^{15} x^6+240699600 a^{14} b x^{13/2}+1564547400 a^{13} b^2 x^7+6327725040 a^{12} b^3 x^{15/2}+17796726675 a^{11} b^4 x^8+36849692880 a^{10} b^5 x^{17/2}+58004146200 a^9 b^6 x^9+70651666800 a^8 b^7 x^{19/2}+67119083460 a^7 b^8 x^{10}+49717839600 a^6 b^9 x^{21/2}+28474762680 a^5 b^{10} x^{11}+12380331600 a^4 b^{11} x^{23/2}+3954828150 a^3 b^{12} x^{12}+876146544 a^2 b^{13} x^{25/2}+120349800 a b^{14} x^{13}+7726160 b^{15} x^{27/2}}{104303160} \]
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Time = 3.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {2 b^{15} x^{\frac {27}{2}}}{27}+\frac {15 a \,b^{14} x^{13}}{13}+\frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5}+\frac {455 a^{3} b^{12} x^{12}}{12}+\frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23}+273 a^{5} b^{10} x^{11}+\frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3}+\frac {1287 a^{7} b^{8} x^{10}}{2}+\frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19}+\frac {5005 a^{9} b^{6} x^{9}}{9}+\frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17}+\frac {1365 a^{11} b^{4} x^{8}}{8}+\frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3}+15 a^{13} b^{2} x^{7}+\frac {30 a^{14} b \,x^{\frac {13}{2}}}{13}+\frac {a^{15} x^{6}}{6}\) | \(168\) |
default | \(\frac {2 b^{15} x^{\frac {27}{2}}}{27}+\frac {15 a \,b^{14} x^{13}}{13}+\frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5}+\frac {455 a^{3} b^{12} x^{12}}{12}+\frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23}+273 a^{5} b^{10} x^{11}+\frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3}+\frac {1287 a^{7} b^{8} x^{10}}{2}+\frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19}+\frac {5005 a^{9} b^{6} x^{9}}{9}+\frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17}+\frac {1365 a^{11} b^{4} x^{8}}{8}+\frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3}+15 a^{13} b^{2} x^{7}+\frac {30 a^{14} b \,x^{\frac {13}{2}}}{13}+\frac {a^{15} x^{6}}{6}\) | \(168\) |
trager | \(\frac {a \left (1080 b^{14} x^{12}+35490 a^{2} b^{12} x^{11}+1080 b^{14} x^{11}+255528 a^{4} b^{10} x^{10}+35490 a^{2} b^{12} x^{10}+1080 b^{14} x^{10}+602316 a^{6} b^{8} x^{9}+255528 a^{4} b^{10} x^{9}+35490 a^{2} b^{12} x^{9}+1080 b^{14} x^{9}+520520 a^{8} b^{6} x^{8}+602316 a^{6} b^{8} x^{8}+255528 a^{4} b^{10} x^{8}+35490 a^{2} b^{12} x^{8}+1080 b^{14} x^{8}+159705 a^{10} b^{4} x^{7}+520520 a^{8} b^{6} x^{7}+602316 a^{6} b^{8} x^{7}+255528 a^{4} b^{10} x^{7}+35490 a^{2} b^{12} x^{7}+1080 x^{7} b^{14}+14040 a^{12} b^{2} x^{6}+159705 a^{10} b^{4} x^{6}+520520 a^{8} b^{6} x^{6}+602316 a^{6} b^{8} x^{6}+255528 a^{4} b^{10} x^{6}+35490 a^{2} b^{12} x^{6}+1080 b^{14} x^{6}+156 a^{14} x^{5}+14040 a^{12} b^{2} x^{5}+159705 a^{10} b^{4} x^{5}+520520 a^{8} b^{6} x^{5}+602316 a^{6} b^{8} x^{5}+255528 a^{4} b^{10} x^{5}+35490 a^{2} b^{12} x^{5}+1080 b^{14} x^{5}+156 x^{4} a^{14}+14040 x^{4} a^{12} b^{2}+159705 x^{4} a^{10} b^{4}+520520 a^{8} b^{6} x^{4}+602316 a^{6} b^{8} x^{4}+255528 x^{4} a^{4} b^{10}+35490 x^{4} a^{2} b^{12}+1080 b^{14} x^{4}+156 a^{14} x^{3}+14040 a^{12} b^{2} x^{3}+159705 a^{10} b^{4} x^{3}+520520 a^{8} b^{6} x^{3}+602316 a^{6} b^{8} x^{3}+255528 a^{4} b^{10} x^{3}+35490 a^{2} b^{12} x^{3}+1080 b^{14} x^{3}+156 a^{14} x^{2}+14040 a^{12} b^{2} x^{2}+159705 a^{10} b^{4} x^{2}+520520 a^{8} b^{6} x^{2}+602316 a^{6} b^{8} x^{2}+255528 a^{4} b^{10} x^{2}+35490 a^{2} b^{12} x^{2}+1080 b^{14} x^{2}+156 a^{14} x +14040 a^{12} b^{2} x +159705 a^{10} b^{4} x +520520 a^{8} b^{6} x +602316 a^{6} b^{8} x +255528 a^{4} b^{10} x +35490 a^{2} b^{12} x +1080 b^{14} x +156 a^{14}+14040 a^{12} b^{2}+159705 a^{10} b^{4}+520520 a^{8} b^{6}+602316 a^{6} b^{8}+255528 a^{4} b^{10}+35490 a^{2} b^{12}+1080 b^{14}\right ) \left (-1+x \right )}{936}+\frac {2 b \,x^{\frac {13}{2}} \left (482885 x^{7} b^{14}+54759159 a^{2} b^{12} x^{6}+773770725 a^{4} b^{10} x^{5}+3107364975 a^{6} b^{8} x^{4}+4415729175 a^{8} b^{6} x^{3}+2303105805 a^{10} b^{4} x^{2}+395482815 a^{12} b^{2} x +15043725 a^{14}\right )}{13037895}\) | \(832\) |
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Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {15}{13} \, a b^{14} x^{13} + \frac {455}{12} \, a^{3} b^{12} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1287}{2} \, a^{7} b^{8} x^{10} + \frac {5005}{9} \, a^{9} b^{6} x^{9} + \frac {1365}{8} \, a^{11} b^{4} x^{8} + 15 \, a^{13} b^{2} x^{7} + \frac {1}{6} \, a^{15} x^{6} + \frac {2}{13037895} \, {\left (482885 \, b^{15} x^{13} + 54759159 \, a^{2} b^{13} x^{12} + 773770725 \, a^{4} b^{11} x^{11} + 3107364975 \, a^{6} b^{9} x^{10} + 4415729175 \, a^{8} b^{7} x^{9} + 2303105805 \, a^{10} b^{5} x^{8} + 395482815 \, a^{12} b^{3} x^{7} + 15043725 \, a^{14} b x^{6}\right )} \sqrt {x} \]
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Time = 1.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {a^{15} x^{6}}{6} + \frac {30 a^{14} b x^{\frac {13}{2}}}{13} + 15 a^{13} b^{2} x^{7} + \frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3} + \frac {1365 a^{11} b^{4} x^{8}}{8} + \frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17} + \frac {5005 a^{9} b^{6} x^{9}}{9} + \frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19} + \frac {1287 a^{7} b^{8} x^{10}}{2} + \frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3} + 273 a^{5} b^{10} x^{11} + \frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23} + \frac {455 a^{3} b^{12} x^{12}}{12} + \frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5} + \frac {15 a b^{14} x^{13}}{13} + \frac {2 b^{15} x^{\frac {27}{2}}}{27} \]
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Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.83 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{27}}{27 \, b^{12}} - \frac {11 \, {\left (b \sqrt {x} + a\right )}^{26} a}{13 \, b^{12}} + \frac {22 \, {\left (b \sqrt {x} + a\right )}^{25} a^{2}}{5 \, b^{12}} - \frac {55 \, {\left (b \sqrt {x} + a\right )}^{24} a^{3}}{4 \, b^{12}} + \frac {660 \, {\left (b \sqrt {x} + a\right )}^{23} a^{4}}{23 \, b^{12}} - \frac {42 \, {\left (b \sqrt {x} + a\right )}^{22} a^{5}}{b^{12}} + \frac {44 \, {\left (b \sqrt {x} + a\right )}^{21} a^{6}}{b^{12}} - \frac {33 \, {\left (b \sqrt {x} + a\right )}^{20} a^{7}}{b^{12}} + \frac {330 \, {\left (b \sqrt {x} + a\right )}^{19} a^{8}}{19 \, b^{12}} - \frac {55 \, {\left (b \sqrt {x} + a\right )}^{18} a^{9}}{9 \, b^{12}} + \frac {22 \, {\left (b \sqrt {x} + a\right )}^{17} a^{10}}{17 \, b^{12}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{11}}{8 \, b^{12}} \]
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {2}{27} \, b^{15} x^{\frac {27}{2}} + \frac {15}{13} \, a b^{14} x^{13} + \frac {42}{5} \, a^{2} b^{13} x^{\frac {25}{2}} + \frac {455}{12} \, a^{3} b^{12} x^{12} + \frac {2730}{23} \, a^{4} b^{11} x^{\frac {23}{2}} + 273 \, a^{5} b^{10} x^{11} + \frac {1430}{3} \, a^{6} b^{9} x^{\frac {21}{2}} + \frac {1287}{2} \, a^{7} b^{8} x^{10} + \frac {12870}{19} \, a^{8} b^{7} x^{\frac {19}{2}} + \frac {5005}{9} \, a^{9} b^{6} x^{9} + \frac {6006}{17} \, a^{10} b^{5} x^{\frac {17}{2}} + \frac {1365}{8} \, a^{11} b^{4} x^{8} + \frac {182}{3} \, a^{12} b^{3} x^{\frac {15}{2}} + 15 \, a^{13} b^{2} x^{7} + \frac {30}{13} \, a^{14} b x^{\frac {13}{2}} + \frac {1}{6} \, a^{15} x^{6} \]
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Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {a^{15}\,x^6}{6}+\frac {2\,b^{15}\,x^{27/2}}{27}+\frac {15\,a\,b^{14}\,x^{13}}{13}+\frac {30\,a^{14}\,b\,x^{13/2}}{13}+15\,a^{13}\,b^2\,x^7+\frac {1365\,a^{11}\,b^4\,x^8}{8}+\frac {5005\,a^9\,b^6\,x^9}{9}+\frac {1287\,a^7\,b^8\,x^{10}}{2}+273\,a^5\,b^{10}\,x^{11}+\frac {455\,a^3\,b^{12}\,x^{12}}{12}+\frac {182\,a^{12}\,b^3\,x^{15/2}}{3}+\frac {6006\,a^{10}\,b^5\,x^{17/2}}{17}+\frac {12870\,a^8\,b^7\,x^{19/2}}{19}+\frac {1430\,a^6\,b^9\,x^{21/2}}{3}+\frac {2730\,a^4\,b^{11}\,x^{23/2}}{23}+\frac {42\,a^2\,b^{13}\,x^{25/2}}{5} \]
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