Integrand size = 13, antiderivative size = 53 \[ \int \frac {x^{3/2}}{a+b x} \, dx=-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {52, 65, 211} \[ \int \frac {x^{3/2}}{a+b x} \, dx=\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b} \]
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Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x} \, dx}{b} \\ & = -\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {a^2 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^2} \\ & = -\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = -\frac {2 a \sqrt {x}}{b^2}+\frac {2 x^{3/2}}{3 b}+\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\frac {2 \sqrt {x} (-3 a+b x)}{3 b^2}+\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {2 \left (-b x +3 a \right ) \sqrt {x}}{3 b^{2}}+\frac {2 a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(42\) |
derivativedivides | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+a \sqrt {x}\right )}{b^{2}}+\frac {2 a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(43\) |
default | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+a \sqrt {x}\right )}{b^{2}}+\frac {2 a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(43\) |
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Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.94 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\left [\frac {3 \, a \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (b x - 3 \, a\right )} \sqrt {x}}{3 \, b^{2}}, \frac {2 \, {\left (3 \, a \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (b x - 3 \, a\right )} \sqrt {x}\right )}}{3 \, b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).
Time = 1.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.02 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: a = 0 \\\frac {a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{3} \sqrt {- \frac {a}{b}}} - \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{3} \sqrt {- \frac {a}{b}}} - \frac {2 a \sqrt {x}}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (b x^{\frac {3}{2}} - 3 \, a \sqrt {x}\right )}}{3 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (b^{2} x^{\frac {3}{2}} - 3 \, a b \sqrt {x}\right )}}{3 \, b^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {x^{3/2}}{a+b x} \, dx=\frac {2\,x^{3/2}}{3\,b}-\frac {2\,a\,\sqrt {x}}{b^2}+\frac {2\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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