Integrand size = 15, antiderivative size = 205 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=30 a^{14} b \sqrt {x}+105 a^{13} b^2 x+\frac {910}{3} a^{12} b^3 x^{3/2}+\frac {1365}{2} a^{11} b^4 x^2+\frac {6006}{5} a^{10} b^5 x^{5/2}+\frac {5005}{3} a^9 b^6 x^3+\frac {12870}{7} a^8 b^7 x^{7/2}+\frac {6435}{4} a^7 b^8 x^4+\frac {10010}{9} a^6 b^9 x^{9/2}+\frac {3003}{5} a^5 b^{10} x^5+\frac {2730}{11} a^4 b^{11} x^{11/2}+\frac {455}{6} a^3 b^{12} x^6+\frac {210}{13} a^2 b^{13} x^{13/2}+\frac {15}{7} a b^{14} x^7+\frac {2}{15} b^{15} x^{15/2}+a^{15} \log (x) \]
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Time = 0.09 (sec) , antiderivative size = 205, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=a^{15} \log (x)+30 a^{14} b \sqrt {x}+105 a^{13} b^2 x+\frac {910}{3} a^{12} b^3 x^{3/2}+\frac {1365}{2} a^{11} b^4 x^2+\frac {6006}{5} a^{10} b^5 x^{5/2}+\frac {5005}{3} a^9 b^6 x^3+\frac {12870}{7} a^8 b^7 x^{7/2}+\frac {6435}{4} a^7 b^8 x^4+\frac {10010}{9} a^6 b^9 x^{9/2}+\frac {3003}{5} a^5 b^{10} x^5+\frac {2730}{11} a^4 b^{11} x^{11/2}+\frac {455}{6} a^3 b^{12} x^6+\frac {210}{13} a^2 b^{13} x^{13/2}+\frac {15}{7} a b^{14} x^7+\frac {2}{15} b^{15} x^{15/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (15 a^{14} b+\frac {a^{15}}{x}+105 a^{13} b^2 x+455 a^{12} b^3 x^2+1365 a^{11} b^4 x^3+3003 a^{10} b^5 x^4+5005 a^9 b^6 x^5+6435 a^8 b^7 x^6+6435 a^7 b^8 x^7+5005 a^6 b^9 x^8+3003 a^5 b^{10} x^9+1365 a^4 b^{11} x^{10}+455 a^3 b^{12} x^{11}+105 a^2 b^{13} x^{12}+15 a b^{14} x^{13}+b^{15} x^{14}\right ) \, dx,x,\sqrt {x}\right ) \\ & = 30 a^{14} b \sqrt {x}+105 a^{13} b^2 x+\frac {910}{3} a^{12} b^3 x^{3/2}+\frac {1365}{2} a^{11} b^4 x^2+\frac {6006}{5} a^{10} b^5 x^{5/2}+\frac {5005}{3} a^9 b^6 x^3+\frac {12870}{7} a^8 b^7 x^{7/2}+\frac {6435}{4} a^7 b^8 x^4+\frac {10010}{9} a^6 b^9 x^{9/2}+\frac {3003}{5} a^5 b^{10} x^5+\frac {2730}{11} a^4 b^{11} x^{11/2}+\frac {455}{6} a^3 b^{12} x^6+\frac {210}{13} a^2 b^{13} x^{13/2}+\frac {15}{7} a b^{14} x^7+\frac {2}{15} b^{15} x^{15/2}+a^{15} \log (x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {5405400 a^{14} b \sqrt {x}+18918900 a^{13} b^2 x+54654600 a^{12} b^3 x^{3/2}+122972850 a^{11} b^4 x^2+216432216 a^{10} b^5 x^{5/2}+300600300 a^9 b^6 x^3+331273800 a^8 b^7 x^{7/2}+289864575 a^7 b^8 x^4+200400200 a^6 b^9 x^{9/2}+108216108 a^5 b^{10} x^5+44717400 a^4 b^{11} x^{11/2}+13663650 a^3 b^{12} x^6+2910600 a^2 b^{13} x^{13/2}+386100 a b^{14} x^7+24024 b^{15} x^{15/2}}{180180}+2 a^{15} \log \left (\sqrt {x}\right ) \]
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Time = 3.54 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(105 a^{13} b^{2} x +\frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{2}}{2}+\frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5}+\frac {5005 a^{9} b^{6} x^{3}}{3}+\frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7}+\frac {6435 a^{7} b^{8} x^{4}}{4}+\frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9}+\frac {3003 a^{5} b^{10} x^{5}}{5}+\frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13}+\frac {15 a \,b^{14} x^{7}}{7}+\frac {2 b^{15} x^{\frac {15}{2}}}{15}+a^{15} \ln \left (x \right )+30 a^{14} b \sqrt {x}\) | \(164\) |
default | \(105 a^{13} b^{2} x +\frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{2}}{2}+\frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5}+\frac {5005 a^{9} b^{6} x^{3}}{3}+\frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7}+\frac {6435 a^{7} b^{8} x^{4}}{4}+\frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9}+\frac {3003 a^{5} b^{10} x^{5}}{5}+\frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13}+\frac {15 a \,b^{14} x^{7}}{7}+\frac {2 b^{15} x^{\frac {15}{2}}}{15}+a^{15} \ln \left (x \right )+30 a^{14} b \sqrt {x}\) | \(164\) |
trager | \(\frac {a \,b^{2} \left (900 b^{12} x^{6}+31850 a^{2} b^{10} x^{5}+900 b^{12} x^{5}+252252 a^{4} b^{8} x^{4}+31850 a^{2} b^{10} x^{4}+900 b^{12} x^{4}+675675 a^{6} b^{6} x^{3}+252252 a^{4} b^{8} x^{3}+31850 a^{2} b^{10} x^{3}+900 b^{12} x^{3}+700700 a^{8} b^{4} x^{2}+675675 a^{6} b^{6} x^{2}+252252 a^{4} b^{8} x^{2}+31850 a^{2} b^{10} x^{2}+900 b^{12} x^{2}+286650 a^{10} b^{2} x +700700 a^{8} b^{4} x +675675 a^{6} b^{6} x +252252 a^{4} b^{8} x +31850 a^{2} b^{10} x +900 b^{12} x +44100 a^{12}+286650 a^{10} b^{2}+700700 a^{8} b^{4}+675675 a^{6} b^{6}+252252 a^{4} b^{8}+31850 a^{2} b^{10}+900 b^{12}\right ) \left (-1+x \right )}{420}+\frac {2 b \left (3003 x^{7} b^{14}+363825 a^{2} b^{12} x^{6}+5589675 a^{4} b^{10} x^{5}+25050025 a^{6} b^{8} x^{4}+41409225 a^{8} b^{6} x^{3}+27054027 a^{10} b^{4} x^{2}+6831825 a^{12} b^{2} x +675675 a^{14}\right ) \sqrt {x}}{45045}+a^{15} \ln \left (x \right )\) | \(353\) |
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Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {15}{7} \, a b^{14} x^{7} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + 105 \, a^{13} b^{2} x + 2 \, a^{15} \log \left (\sqrt {x}\right ) + \frac {2}{45045} \, {\left (3003 \, b^{15} x^{7} + 363825 \, a^{2} b^{13} x^{6} + 5589675 \, a^{4} b^{11} x^{5} + 25050025 \, a^{6} b^{9} x^{4} + 41409225 \, a^{8} b^{7} x^{3} + 27054027 \, a^{10} b^{5} x^{2} + 6831825 \, a^{12} b^{3} x + 675675 \, a^{14} b\right )} \sqrt {x} \]
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Time = 0.46 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=a^{15} \log {\left (x \right )} + 30 a^{14} b \sqrt {x} + 105 a^{13} b^{2} x + \frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3} + \frac {1365 a^{11} b^{4} x^{2}}{2} + \frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5} + \frac {5005 a^{9} b^{6} x^{3}}{3} + \frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7} + \frac {6435 a^{7} b^{8} x^{4}}{4} + \frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9} + \frac {3003 a^{5} b^{10} x^{5}}{5} + \frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11} + \frac {455 a^{3} b^{12} x^{6}}{6} + \frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13} + \frac {15 a b^{14} x^{7}}{7} + \frac {2 b^{15} x^{\frac {15}{2}}}{15} \]
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Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {2}{15} \, b^{15} x^{\frac {15}{2}} + \frac {15}{7} \, a b^{14} x^{7} + \frac {210}{13} \, a^{2} b^{13} x^{\frac {13}{2}} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {2730}{11} \, a^{4} b^{11} x^{\frac {11}{2}} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {10010}{9} \, a^{6} b^{9} x^{\frac {9}{2}} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {12870}{7} \, a^{8} b^{7} x^{\frac {7}{2}} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {6006}{5} \, a^{10} b^{5} x^{\frac {5}{2}} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + \frac {910}{3} \, a^{12} b^{3} x^{\frac {3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left (x\right ) + 30 \, a^{14} b \sqrt {x} \]
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Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {2}{15} \, b^{15} x^{\frac {15}{2}} + \frac {15}{7} \, a b^{14} x^{7} + \frac {210}{13} \, a^{2} b^{13} x^{\frac {13}{2}} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {2730}{11} \, a^{4} b^{11} x^{\frac {11}{2}} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {10010}{9} \, a^{6} b^{9} x^{\frac {9}{2}} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {12870}{7} \, a^{8} b^{7} x^{\frac {7}{2}} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {6006}{5} \, a^{10} b^{5} x^{\frac {5}{2}} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + \frac {910}{3} \, a^{12} b^{3} x^{\frac {3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left ({\left | x \right |}\right ) + 30 \, a^{14} b \sqrt {x} \]
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Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=2\,a^{15}\,\ln \left (\sqrt {x}\right )+\frac {2\,b^{15}\,x^{15/2}}{15}+105\,a^{13}\,b^2\,x+30\,a^{14}\,b\,\sqrt {x}+\frac {15\,a\,b^{14}\,x^7}{7}+\frac {1365\,a^{11}\,b^4\,x^2}{2}+\frac {5005\,a^9\,b^6\,x^3}{3}+\frac {6435\,a^7\,b^8\,x^4}{4}+\frac {3003\,a^5\,b^{10}\,x^5}{5}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{6}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{5}+\frac {12870\,a^8\,b^7\,x^{7/2}}{7}+\frac {10010\,a^6\,b^9\,x^{9/2}}{9}+\frac {2730\,a^4\,b^{11}\,x^{11/2}}{11}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{13} \]
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