Integrand size = 15, antiderivative size = 130 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {20 a^9 b}{9 x^{9/2}}-\frac {45 a^8 b^2}{4 x^4}-\frac {240 a^7 b^3}{7 x^{7/2}}-\frac {70 a^6 b^4}{x^3}-\frac {504 a^5 b^5}{5 x^{5/2}}-\frac {105 a^4 b^6}{x^2}-\frac {80 a^3 b^7}{x^{3/2}}-\frac {45 a^2 b^8}{x}-\frac {20 a b^9}{\sqrt {x}}+b^{10} \log (x) \]
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Time = 0.05 (sec) , antiderivative size = 130, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {20 a^9 b}{9 x^{9/2}}-\frac {45 a^8 b^2}{4 x^4}-\frac {240 a^7 b^3}{7 x^{7/2}}-\frac {70 a^6 b^4}{x^3}-\frac {504 a^5 b^5}{5 x^{5/2}}-\frac {105 a^4 b^6}{x^2}-\frac {80 a^3 b^7}{x^{3/2}}-\frac {45 a^2 b^8}{x}-\frac {20 a b^9}{\sqrt {x}}+b^{10} \log (x) \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{11}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^{10}}{x^{11}}+\frac {10 a^9 b}{x^{10}}+\frac {45 a^8 b^2}{x^9}+\frac {120 a^7 b^3}{x^8}+\frac {210 a^6 b^4}{x^7}+\frac {252 a^5 b^5}{x^6}+\frac {210 a^4 b^6}{x^5}+\frac {120 a^3 b^7}{x^4}+\frac {45 a^2 b^8}{x^3}+\frac {10 a b^9}{x^2}+\frac {b^{10}}{x}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{10}}{5 x^5}-\frac {20 a^9 b}{9 x^{9/2}}-\frac {45 a^8 b^2}{4 x^4}-\frac {240 a^7 b^3}{7 x^{7/2}}-\frac {70 a^6 b^4}{x^3}-\frac {504 a^5 b^5}{5 x^{5/2}}-\frac {105 a^4 b^6}{x^2}-\frac {80 a^3 b^7}{x^{3/2}}-\frac {45 a^2 b^8}{x}-\frac {20 a b^9}{\sqrt {x}}+b^{10} \log (x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=\frac {-252 a^{10}-2800 a^9 b \sqrt {x}-14175 a^8 b^2 x-43200 a^7 b^3 x^{3/2}-88200 a^6 b^4 x^2-127008 a^5 b^5 x^{5/2}-132300 a^4 b^6 x^3-100800 a^3 b^7 x^{7/2}-56700 a^2 b^8 x^4-25200 a b^9 x^{9/2}}{1260 x^5}+2 b^{10} \log \left (\sqrt {x}\right ) \]
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Time = 3.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {a^{10}}{5 x^{5}}-\frac {20 a^{9} b}{9 x^{\frac {9}{2}}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}}-\frac {45 a^{2} b^{8}}{x}+b^{10} \ln \left (x \right )-\frac {20 a \,b^{9}}{\sqrt {x}}\) | \(111\) |
default | \(-\frac {a^{10}}{5 x^{5}}-\frac {20 a^{9} b}{9 x^{\frac {9}{2}}}-\frac {45 a^{8} b^{2}}{4 x^{4}}-\frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}}-\frac {70 a^{6} b^{4}}{x^{3}}-\frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}}-\frac {45 a^{2} b^{8}}{x}+b^{10} \ln \left (x \right )-\frac {20 a \,b^{9}}{\sqrt {x}}\) | \(111\) |
trager | \(\frac {\left (-1+x \right ) \left (4 a^{8} x^{4}+225 x^{4} b^{2} a^{6}+1400 a^{4} x^{4} b^{4}+2100 a^{2} b^{6} x^{4}+900 b^{8} x^{4}+4 x^{3} a^{8}+225 a^{6} b^{2} x^{3}+1400 a^{4} b^{4} x^{3}+2100 a^{2} b^{6} x^{3}+4 a^{8} x^{2}+225 a^{6} x^{2} b^{2}+1400 a^{4} b^{4} x^{2}+4 a^{8} x +225 a^{6} b^{2} x +4 a^{8}\right ) a^{2}}{20 x^{5}}-\frac {4 \left (1575 b^{8} x^{4}+6300 a^{2} b^{6} x^{3}+7938 a^{4} b^{4} x^{2}+2700 a^{6} b^{2} x +175 a^{8}\right ) a b}{315 x^{\frac {9}{2}}}-b^{10} \ln \left (\frac {1}{x}\right )\) | \(215\) |
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Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=\frac {2520 \, b^{10} x^{5} \log \left (\sqrt {x}\right ) - 56700 \, a^{2} b^{8} x^{4} - 132300 \, a^{4} b^{6} x^{3} - 88200 \, a^{6} b^{4} x^{2} - 14175 \, a^{8} b^{2} x - 252 \, a^{10} - 16 \, {\left (1575 \, a b^{9} x^{4} + 6300 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 2700 \, a^{7} b^{3} x + 175 \, a^{9} b\right )} \sqrt {x}}{1260 \, x^{5}} \]
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Time = 0.37 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=- \frac {a^{10}}{5 x^{5}} - \frac {20 a^{9} b}{9 x^{\frac {9}{2}}} - \frac {45 a^{8} b^{2}}{4 x^{4}} - \frac {240 a^{7} b^{3}}{7 x^{\frac {7}{2}}} - \frac {70 a^{6} b^{4}}{x^{3}} - \frac {504 a^{5} b^{5}}{5 x^{\frac {5}{2}}} - \frac {105 a^{4} b^{6}}{x^{2}} - \frac {80 a^{3} b^{7}}{x^{\frac {3}{2}}} - \frac {45 a^{2} b^{8}}{x} - \frac {20 a b^{9}}{\sqrt {x}} + b^{10} \log {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=b^{10} \log \left (x\right ) - \frac {25200 \, a b^{9} x^{\frac {9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac {7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac {5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac {3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt {x} + 252 \, a^{10}}{1260 \, x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=b^{10} \log \left ({\left | x \right |}\right ) - \frac {25200 \, a b^{9} x^{\frac {9}{2}} + 56700 \, a^{2} b^{8} x^{4} + 100800 \, a^{3} b^{7} x^{\frac {7}{2}} + 132300 \, a^{4} b^{6} x^{3} + 127008 \, a^{5} b^{5} x^{\frac {5}{2}} + 88200 \, a^{6} b^{4} x^{2} + 43200 \, a^{7} b^{3} x^{\frac {3}{2}} + 14175 \, a^{8} b^{2} x + 2800 \, a^{9} b \sqrt {x} + 252 \, a^{10}}{1260 \, x^{5}} \]
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Time = 5.87 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^6} \, dx=2\,b^{10}\,\ln \left (\sqrt {x}\right )-\frac {\frac {a^{10}}{5}+\frac {45\,a^8\,b^2\,x}{4}+\frac {20\,a^9\,b\,\sqrt {x}}{9}+20\,a\,b^9\,x^{9/2}+70\,a^6\,b^4\,x^2+105\,a^4\,b^6\,x^3+45\,a^2\,b^8\,x^4+\frac {240\,a^7\,b^3\,x^{3/2}}{7}+\frac {504\,a^5\,b^5\,x^{5/2}}{5}+80\,a^3\,b^7\,x^{7/2}}{x^5} \]
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