Integrand size = 15, antiderivative size = 62 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 62, 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} \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+10 a^3 b^2 \log (x)+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (10 a^2 b^3+\frac {a^5}{x^3}+\frac {5 a^4 b}{x^2}+\frac {10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=-\frac {a^5}{x}-\frac {10 a^4 b}{\sqrt {x}}+20 a^2 b^3 \sqrt {x}+5 a b^4 x+\frac {2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x) \]
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Time = 3.59 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {a^{5}}{x}+5 b^{4} x a +\frac {2 b^{5} x^{\frac {3}{2}}}{3}+10 a^{3} b^{2} \ln \left (x \right )-\frac {10 a^{4} b}{\sqrt {x}}+20 a^{2} b^{3} \sqrt {x}\) | \(55\) |
default | \(-\frac {a^{5}}{x}+5 b^{4} x a +\frac {2 b^{5} x^{\frac {3}{2}}}{3}+10 a^{3} b^{2} \ln \left (x \right )-\frac {10 a^{4} b}{\sqrt {x}}+20 a^{2} b^{3} \sqrt {x}\) | \(55\) |
trager | \(\frac {\left (-1+x \right ) \left (5 b^{4} x +a^{4}\right ) a}{x}-\frac {2 \left (-b^{4} x^{2}-30 a^{2} b^{2} x +15 a^{4}\right ) b}{3 \sqrt {x}}+10 a^{3} b^{2} \ln \left (x \right )\) | \(59\) |
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {15 \, a b^{4} x^{2} + 60 \, a^{3} b^{2} x \log \left (\sqrt {x}\right ) - 3 \, a^{5} + 2 \, {\left (b^{5} x^{2} + 30 \, a^{2} b^{3} x - 15 \, a^{4} b\right )} \sqrt {x}}{3 \, x} \]
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Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=- \frac {a^{5}}{x} - \frac {10 a^{4} b}{\sqrt {x}} + 10 a^{3} b^{2} \log {\left (x \right )} + 20 a^{2} b^{3} \sqrt {x} + 5 a b^{4} x + \frac {2 b^{5} x^{\frac {3}{2}}}{3} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2}{3} \, b^{5} x^{\frac {3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left (x\right ) + 20 \, a^{2} b^{3} \sqrt {x} - \frac {10 \, a^{4} b \sqrt {x} + a^{5}}{x} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2}{3} \, b^{5} x^{\frac {3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) + 20 \, a^{2} b^{3} \sqrt {x} - \frac {10 \, a^{4} b \sqrt {x} + a^{5}}{x} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^5}{x^2} \, dx=\frac {2\,b^5\,x^{3/2}}{3}-\frac {a^5+10\,a^4\,b\,\sqrt {x}}{x}+20\,a^3\,b^2\,\ln \left (\sqrt {x}\right )+20\,a^2\,b^3\,\sqrt {x}+5\,a\,b^4\,x \]
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