Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+20 a b^9 \sqrt {x}+b^{10} x+45 a^2 b^8 \log (x) \]
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Time = 0.05 (sec) , antiderivative size = 122, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=-\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+45 a^2 b^8 \log (x)+20 a b^9 \sqrt {x}+b^{10} x \]
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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^9} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (10 a b^9+\frac {a^{10}}{x^9}+\frac {10 a^9 b}{x^8}+\frac {45 a^8 b^2}{x^7}+\frac {120 a^7 b^3}{x^6}+\frac {210 a^6 b^4}{x^5}+\frac {252 a^5 b^5}{x^4}+\frac {210 a^4 b^6}{x^3}+\frac {120 a^3 b^7}{x^2}+\frac {45 a^2 b^8}{x}+b^{10} x\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{10}}{4 x^4}-\frac {20 a^9 b}{7 x^{7/2}}-\frac {15 a^8 b^2}{x^3}-\frac {48 a^7 b^3}{x^{5/2}}-\frac {105 a^6 b^4}{x^2}-\frac {168 a^5 b^5}{x^{3/2}}-\frac {210 a^4 b^6}{x}-\frac {240 a^3 b^7}{\sqrt {x}}+20 a b^9 \sqrt {x}+b^{10} x+45 a^2 b^8 \log (x) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=-\frac {7 a^{10}+80 a^9 b \sqrt {x}+420 a^8 b^2 x+1344 a^7 b^3 x^{3/2}+2940 a^6 b^4 x^2+4704 a^5 b^5 x^{5/2}+5880 a^4 b^6 x^3+6720 a^3 b^7 x^{7/2}-560 a b^9 x^{9/2}-28 b^{10} x^5}{28 x^4}+45 a^2 b^8 \log (x) \]
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Time = 3.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {a^{10}}{4 x^{4}}-\frac {20 a^{9} b}{7 x^{\frac {7}{2}}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}}-\frac {210 a^{4} b^{6}}{x}+b^{10} x +45 a^{2} b^{8} \ln \left (x \right )-\frac {240 a^{3} b^{7}}{\sqrt {x}}+20 a \,b^{9} \sqrt {x}\) | \(109\) |
default | \(-\frac {a^{10}}{4 x^{4}}-\frac {20 a^{9} b}{7 x^{\frac {7}{2}}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}}-\frac {105 a^{6} b^{4}}{x^{2}}-\frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}}-\frac {210 a^{4} b^{6}}{x}+b^{10} x +45 a^{2} b^{8} \ln \left (x \right )-\frac {240 a^{3} b^{7}}{\sqrt {x}}+20 a \,b^{9} \sqrt {x}\) | \(109\) |
trager | \(\frac {\left (-1+x \right ) \left (4 b^{10} x^{4}+a^{10} x^{3}+60 a^{8} b^{2} x^{3}+420 a^{6} b^{4} x^{3}+840 a^{4} b^{6} x^{3}+a^{10} x^{2}+60 a^{8} b^{2} x^{2}+420 x^{2} a^{6} b^{4}+a^{10} x +60 a^{8} b^{2} x +a^{10}\right )}{4 x^{4}}-\frac {4 \left (-35 b^{8} x^{4}+420 a^{2} b^{6} x^{3}+294 a^{4} b^{4} x^{2}+84 a^{6} b^{2} x +5 a^{8}\right ) a b}{7 x^{\frac {7}{2}}}-45 a^{2} b^{8} \ln \left (\frac {1}{x}\right )\) | \(169\) |
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Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=\frac {28 \, b^{10} x^{5} + 2520 \, a^{2} b^{8} x^{4} \log \left (\sqrt {x}\right ) - 5880 \, a^{4} b^{6} x^{3} - 2940 \, a^{6} b^{4} x^{2} - 420 \, a^{8} b^{2} x - 7 \, a^{10} + 16 \, {\left (35 \, a b^{9} x^{4} - 420 \, a^{3} b^{7} x^{3} - 294 \, a^{5} b^{5} x^{2} - 84 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt {x}}{28 \, x^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=- \frac {a^{10}}{4 x^{4}} - \frac {20 a^{9} b}{7 x^{\frac {7}{2}}} - \frac {15 a^{8} b^{2}}{x^{3}} - \frac {48 a^{7} b^{3}}{x^{\frac {5}{2}}} - \frac {105 a^{6} b^{4}}{x^{2}} - \frac {168 a^{5} b^{5}}{x^{\frac {3}{2}}} - \frac {210 a^{4} b^{6}}{x} - \frac {240 a^{3} b^{7}}{\sqrt {x}} + 45 a^{2} b^{8} \log {\left (x \right )} + 20 a b^{9} \sqrt {x} + b^{10} x \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10} x + 45 \, a^{2} b^{8} \log \left (x\right ) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) + 20 \, a b^{9} \sqrt {x} - \frac {6720 \, a^{3} b^{7} x^{\frac {7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac {5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac {3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt {x} + 7 \, a^{10}}{28 \, x^{4}} \]
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Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^5} \, dx=b^{10}\,x-\frac {\frac {a^{10}}{4}+15\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{7}+105\,a^6\,b^4\,x^2+210\,a^4\,b^6\,x^3+48\,a^7\,b^3\,x^{3/2}+168\,a^5\,b^5\,x^{5/2}+240\,a^3\,b^7\,x^{7/2}}{x^4}+90\,a^2\,b^8\,\ln \left (\sqrt {x}\right )+20\,a\,b^9\,\sqrt {x} \]
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