Integrand size = 18, antiderivative size = 69 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {2 \sqrt {a} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {81, 52, 65, 211} \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=-\frac {2 \sqrt {a} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \sqrt {x} (A b-a B)}{b^2}+\frac {2 B x^{3/2}}{3 b} \]
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Rule 52
Rule 65
Rule 81
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2 B x^{3/2}}{3 b}+\frac {\left (2 \left (\frac {3 A b}{2}-\frac {3 a B}{2}\right )\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{3 b} \\ & = \frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {(a (A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^2} \\ & = \frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {(2 a (A b-a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\frac {2 \sqrt {x} (3 A b-3 a B+b B x)}{3 b^2}+\frac {2 \sqrt {a} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.46 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {2 \left (b B x +3 A b -3 B a \right ) \sqrt {x}}{3 b^{2}}-\frac {2 a \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-2 B a \sqrt {x}}{b^{2}}-\frac {2 a \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(58\) |
default | \(\frac {\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-2 B a \sqrt {x}}{b^{2}}-\frac {2 a \left (A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(58\) |
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Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.87 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\left [-\frac {3 \, {\left (B a - A b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}}{3 \, b^{2}}, \frac {2 \, {\left (3 \, {\left (B a - A b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}\right )}}{3 \, b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (66) = 132\).
Time = 0.79 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.20 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{b} & \text {for}\: a = 0 \\- \frac {A a \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{2} \sqrt {- \frac {a}{b}}} + \frac {A a \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{2} \sqrt {- \frac {a}{b}}} + \frac {2 A \sqrt {x}}{b} + \frac {B a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{b^{3} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{b^{3} \sqrt {- \frac {a}{b}}} - \frac {2 B a \sqrt {x}}{b^{2}} + \frac {2 B x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (B b^{2} x^{\frac {3}{2}} - 3 \, B a b \sqrt {x} + 3 \, A b^{2} \sqrt {x}\right )}}{3 \, b^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {x} (A+B x)}{a+b x} \, dx=\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )+\frac {2\,B\,x^{3/2}}{3\,b}+\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^2-A\,a\,b}\right )\,\left (A\,b-B\,a\right )}{b^{5/2}} \]
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