Integrand size = 15, antiderivative size = 192 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=-\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2}+105 a^{13} b^2 \log (x) \]
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Time = 0.09 (sec) , antiderivative size = 192, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=-\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2} \]
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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^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (455 a^{12} b^3+\frac {a^{15}}{x^3}+\frac {15 a^{14} b}{x^2}+\frac {105 a^{13} b^2}{x}+1365 a^{11} b^4 x+3003 a^{10} b^5 x^2+5005 a^9 b^6 x^3+6435 a^8 b^7 x^4+6435 a^7 b^8 x^5+5005 a^6 b^9 x^6+3003 a^5 b^{10} x^7+1365 a^4 b^{11} x^8+455 a^3 b^{12} x^9+105 a^2 b^{13} x^{10}+15 a b^{14} x^{11}+b^{15} x^{12}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2}+105 a^{13} b^2 \log (x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {-1716 a^{15}-51480 a^{14} b \sqrt {x}+1561560 a^{12} b^3 x^{3/2}+2342340 a^{11} b^4 x^2+3435432 a^{10} b^5 x^{5/2}+4294290 a^9 b^6 x^3+4416984 a^8 b^7 x^{7/2}+3680820 a^7 b^8 x^4+2453880 a^6 b^9 x^{9/2}+1288287 a^5 b^{10} x^5+520520 a^4 b^{11} x^{11/2}+156156 a^3 b^{12} x^6+32760 a^2 b^{13} x^{13/2}+4290 a b^{14} x^7+264 b^{15} x^{15/2}}{1716 x}+210 a^{13} b^2 \log \left (\sqrt {x}\right ) \]
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Time = 3.59 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {a^{15}}{x}+1365 a^{11} b^{4} x +2002 a^{10} b^{5} x^{\frac {3}{2}}+\frac {5005 a^{9} b^{6} x^{2}}{2}+2574 a^{8} b^{7} x^{\frac {5}{2}}+2145 a^{7} b^{8} x^{3}+1430 a^{6} b^{9} x^{\frac {7}{2}}+\frac {3003 a^{5} b^{10} x^{4}}{4}+\frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3}+91 a^{3} b^{12} x^{5}+\frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{14} x^{6}}{2}+\frac {2 b^{15} x^{\frac {13}{2}}}{13}+105 a^{13} b^{2} \ln \left (x \right )-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^{3} \sqrt {x}\) | \(165\) |
default | \(-\frac {a^{15}}{x}+1365 a^{11} b^{4} x +2002 a^{10} b^{5} x^{\frac {3}{2}}+\frac {5005 a^{9} b^{6} x^{2}}{2}+2574 a^{8} b^{7} x^{\frac {5}{2}}+2145 a^{7} b^{8} x^{3}+1430 a^{6} b^{9} x^{\frac {7}{2}}+\frac {3003 a^{5} b^{10} x^{4}}{4}+\frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3}+91 a^{3} b^{12} x^{5}+\frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{14} x^{6}}{2}+\frac {2 b^{15} x^{\frac {13}{2}}}{13}+105 a^{13} b^{2} \ln \left (x \right )-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^{3} \sqrt {x}\) | \(165\) |
trager | \(\frac {\left (-1+x \right ) \left (10 b^{14} x^{6}+364 a^{2} b^{12} x^{5}+10 b^{14} x^{5}+3003 x^{4} a^{4} b^{10}+364 x^{4} a^{2} b^{12}+10 b^{14} x^{4}+8580 a^{6} b^{8} x^{3}+3003 a^{4} b^{10} x^{3}+364 a^{2} b^{12} x^{3}+10 b^{14} x^{3}+10010 a^{8} b^{6} x^{2}+8580 a^{6} b^{8} x^{2}+3003 a^{4} b^{10} x^{2}+364 a^{2} b^{12} x^{2}+10 b^{14} x^{2}+5460 a^{10} b^{4} x +10010 a^{8} b^{6} x +8580 a^{6} b^{8} x +3003 a^{4} b^{10} x +364 a^{2} b^{12} x +10 b^{14} x +4 a^{14}\right ) a}{4 x}-\frac {2 \left (-33 x^{7} b^{14}-4095 a^{2} b^{12} x^{6}-65065 a^{4} b^{10} x^{5}-306735 a^{6} b^{8} x^{4}-552123 a^{8} b^{6} x^{3}-429429 a^{10} b^{4} x^{2}-195195 a^{12} b^{2} x +6435 a^{14}\right ) b}{429 \sqrt {x}}+105 a^{13} b^{2} \ln \left (x \right )\) | \(312\) |
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Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {4290 \, a b^{14} x^{7} + 156156 \, a^{3} b^{12} x^{6} + 1288287 \, a^{5} b^{10} x^{5} + 3680820 \, a^{7} b^{8} x^{4} + 4294290 \, a^{9} b^{6} x^{3} + 2342340 \, a^{11} b^{4} x^{2} + 360360 \, a^{13} b^{2} x \log \left (\sqrt {x}\right ) - 1716 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 4095 \, a^{2} b^{13} x^{6} + 65065 \, a^{4} b^{11} x^{5} + 306735 \, a^{6} b^{9} x^{4} + 552123 \, a^{8} b^{7} x^{3} + 429429 \, a^{10} b^{5} x^{2} + 195195 \, a^{12} b^{3} x - 6435 \, a^{14} b\right )} \sqrt {x}}{1716 \, x} \]
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Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=- \frac {a^{15}}{x} - \frac {30 a^{14} b}{\sqrt {x}} + 105 a^{13} b^{2} \log {\left (x \right )} + 910 a^{12} b^{3} \sqrt {x} + 1365 a^{11} b^{4} x + 2002 a^{10} b^{5} x^{\frac {3}{2}} + \frac {5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac {5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac {7}{2}} + \frac {3003 a^{5} b^{10} x^{4}}{4} + \frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11} + \frac {5 a b^{14} x^{6}}{2} + \frac {2 b^{15} x^{\frac {13}{2}}}{13} \]
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Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left (x\right ) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]
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Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left ({\left | x \right |}\right ) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]
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Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx=\frac {2\,b^{15}\,x^{13/2}}{13}-\frac {a^{15}+30\,a^{14}\,b\,\sqrt {x}}{x}+210\,a^{13}\,b^2\,\ln \left (\sqrt {x}\right )+1365\,a^{11}\,b^4\,x+\frac {5\,a\,b^{14}\,x^6}{2}+\frac {5005\,a^9\,b^6\,x^2}{2}+2145\,a^7\,b^8\,x^3+910\,a^{12}\,b^3\,\sqrt {x}+\frac {3003\,a^5\,b^{10}\,x^4}{4}+91\,a^3\,b^{12}\,x^5+2002\,a^{10}\,b^5\,x^{3/2}+2574\,a^8\,b^7\,x^{5/2}+1430\,a^6\,b^9\,x^{7/2}+\frac {910\,a^4\,b^{11}\,x^{9/2}}{3}+\frac {210\,a^2\,b^{13}\,x^{11/2}}{11} \]
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