Integrand size = 13, antiderivative size = 51 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {\sqrt {x} \left (6 a^2-3 a b \sqrt {x}+2 b^2 x\right )}{3 b^3}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \]
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Time = 5.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-a b x +2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{4}}\) | \(44\) |
default | \(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-a b x +2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{4}}\) | \(44\) |
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Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \]
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Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\begin {cases} - \frac {2 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{4}} + \frac {2 a^{2} \sqrt {x}}{b^{3}} - \frac {a x}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{\frac {3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt {x}}{3 \, b^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {2\,x^{3/2}}{3\,b}-\frac {2\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^4}+\frac {2\,a^2\,\sqrt {x}}{b^3}-\frac {a\,x}{b^2} \]
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