Integrand size = 15, antiderivative size = 47 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \]
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Time = 0.01 (sec) , antiderivative size = 47, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^3}{x^{11}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^3}{x^{11}}+\frac {3 a^2 b}{x^{10}}+\frac {3 a b^2}{x^9}+\frac {b^3}{x^8}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^3}{5 x^5}-\frac {2 a^2 b}{3 x^{9/2}}-\frac {3 a b^2}{4 x^4}-\frac {2 b^3}{7 x^{7/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=\frac {-84 a^3-280 a^2 b \sqrt {x}-315 a b^2 x-120 b^3 x^{3/2}}{420 x^5} \]
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Time = 3.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {a^{3}}{5 x^{5}}-\frac {2 a^{2} b}{3 x^{\frac {9}{2}}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {2 b^{3}}{7 x^{\frac {7}{2}}}\) | \(36\) |
default | \(-\frac {a^{3}}{5 x^{5}}-\frac {2 a^{2} b}{3 x^{\frac {9}{2}}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {2 b^{3}}{7 x^{\frac {7}{2}}}\) | \(36\) |
trager | \(\frac {\left (-1+x \right ) \left (4 a^{2} x^{4}+15 b^{2} x^{4}+4 a^{2} x^{3}+15 b^{2} x^{3}+4 a^{2} x^{2}+15 b^{2} x^{2}+4 a^{2} x +15 b^{2} x +4 a^{2}\right ) a}{20 x^{5}}-\frac {2 \left (3 b^{2} x +7 a^{2}\right ) b}{21 x^{\frac {9}{2}}}\) | \(95\) |
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {315 \, a b^{2} x + 84 \, a^{3} + 40 \, {\left (3 \, b^{3} x + 7 \, a^{2} b\right )} \sqrt {x}}{420 \, x^{5}} \]
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Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=- \frac {a^{3}}{5 x^{5}} - \frac {2 a^{2} b}{3 x^{\frac {9}{2}}} - \frac {3 a b^{2}}{4 x^{4}} - \frac {2 b^{3}}{7 x^{\frac {7}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {120 \, b^{3} x^{\frac {3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt {x} + 84 \, a^{3}}{420 \, x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {120 \, b^{3} x^{\frac {3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt {x} + 84 \, a^{3}}{420 \, x^{5}} \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b \sqrt {x}\right )^3}{x^6} \, dx=-\frac {84\,a^3+120\,b^3\,x^{3/2}+280\,a^2\,b\,\sqrt {x}+315\,a\,b^2\,x}{420\,x^5} \]
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