Integrand size = 22, antiderivative size = 69 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {2 \sqrt {b} (b B-A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {795, 81, 52, 65, 211} \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\frac {2 \sqrt {b} (b B-A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}-\frac {2 \sqrt {x} (b B-A c)}{c^2}+\frac {2 B x^{3/2}}{3 c} \]
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Rule 52
Rule 65
Rule 81
Rule 211
Rule 795
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x} (A+B x)}{b+c x} \, dx \\ & = \frac {2 B x^{3/2}}{3 c}+\frac {\left (2 \left (-\frac {3 b B}{2}+\frac {3 A c}{2}\right )\right ) \int \frac {\sqrt {x}}{b+c x} \, dx}{3 c} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {(b (b B-A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{c^2} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {(2 b (b B-A c)) \text {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {2 \sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\frac {2 \sqrt {x} (-3 b B+3 A c+B c x)}{3 c^2}+\frac {2 \sqrt {b} (b B-A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {2 \left (B c x +3 A c -3 B b \right ) \sqrt {x}}{3 c^{2}}-\frac {2 b \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{c^{2} \sqrt {b c}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 B c \,x^{\frac {3}{2}}}{3}+2 A c \sqrt {x}-2 \sqrt {x}\, B b}{c^{2}}-\frac {2 b \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{c^{2} \sqrt {b c}}\) | \(58\) |
default | \(\frac {\frac {2 B c \,x^{\frac {3}{2}}}{3}+2 A c \sqrt {x}-2 \sqrt {x}\, B b}{c^{2}}-\frac {2 b \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{c^{2} \sqrt {b c}}\) | \(58\) |
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Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.87 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\left [-\frac {3 \, {\left (B b - A c\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x - 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (B c x - 3 \, B b + 3 \, A c\right )} \sqrt {x}}{3 \, c^{2}}, \frac {2 \, {\left (3 \, {\left (B b - A c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) + {\left (B c x - 3 \, B b + 3 \, A c\right )} \sqrt {x}\right )}}{3 \, c^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (66) = 132\).
Time = 2.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.20 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{b} & \text {for}\: c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{c} & \text {for}\: b = 0 \\- \frac {A b \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{c^{2} \sqrt {- \frac {b}{c}}} + \frac {A b \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{c^{2} \sqrt {- \frac {b}{c}}} + \frac {2 A \sqrt {x}}{c} + \frac {B b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{c^{3} \sqrt {- \frac {b}{c}}} - \frac {B b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{c^{3} \sqrt {- \frac {b}{c}}} - \frac {2 B b \sqrt {x}}{c^{2}} + \frac {2 B x^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\frac {2 \, {\left (B b^{2} - A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {2 \, {\left (B c x^{\frac {3}{2}} - 3 \, {\left (B b - A c\right )} \sqrt {x}\right )}}{3 \, c^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\frac {2 \, {\left (B b^{2} - A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {2 \, {\left (B c^{2} x^{\frac {3}{2}} - 3 \, B b c \sqrt {x} + 3 \, A c^{2} \sqrt {x}\right )}}{3 \, c^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {x^{3/2} (A+B x)}{b x+c x^2} \, dx=\sqrt {x}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )+\frac {2\,B\,x^{3/2}}{3\,c}+\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,\sqrt {x}\,\left (A\,c-B\,b\right )}{B\,b^2-A\,b\,c}\right )\,\left (A\,c-B\,b\right )}{c^{5/2}} \]
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