Integrand size = 22, antiderivative size = 85 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}} \]
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Time = 0.03 (sec) , antiderivative size = 85, 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, 79, 53, 65, 211} \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {(b B-3 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}}+\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)} \]
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Rule 53
Rule 65
Rule 79
Rule 211
Rule 795
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{x^{3/2} (b+c x)^2} \, dx \\ & = -\frac {b B-A c}{b c \sqrt {x} (b+c x)}-\frac {\left (\frac {b B}{2}-\frac {3 A c}{2}\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{b c} \\ & = \frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b^2} \\ & = \frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \text {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {-2 A b+b B x-3 A c x}{b^2 \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}} \]
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Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (3 A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{2}}-\frac {2 A}{b^{2} \sqrt {x}}\) | \(64\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (3 A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{2}}-\frac {2 A}{b^{2} \sqrt {x}}\) | \(64\) |
risch | \(-\frac {2 A}{b^{2} \sqrt {x}}-\frac {\frac {2 \left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (3 A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}}}{b^{2}}\) | \(64\) |
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Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\left [\frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{2} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b + 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) - 2 \, {\left (2 \, A b^{2} c - {\left (B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{2} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (2 \, A b^{2} c - {\left (B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{b^{3} c^{2} x^{2} + b^{4} c x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (73) = 146\).
Time = 12.68 (sec) , antiderivative size = 794, normalized size of antiderivative = 9.34 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{b^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c^{2}} & \text {for}\: b = 0 \\- \frac {3 A b c \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} + \frac {3 A b c \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} - \frac {4 A b c \sqrt {- \frac {b}{c}}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} - \frac {3 A c^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} + \frac {3 A c^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} - \frac {6 A c^{2} x \sqrt {- \frac {b}{c}}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} + \frac {B b^{2} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} - \frac {B b^{2} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} + \frac {B b c x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} - \frac {B b c x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} + \frac {2 B b c x \sqrt {- \frac {b}{c}}}{2 b^{3} c \sqrt {x} \sqrt {- \frac {b}{c}} + 2 b^{2} c^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=-\frac {2 \, A b - {\left (B b - 3 \, A c\right )} x}{b^{2} c x^{\frac {3}{2}} + b^{3} \sqrt {x}} + \frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} + \frac {B b x - 3 \, A c x - 2 \, A b}{{\left (c x^{\frac {3}{2}} + b \sqrt {x}\right )} b^{2}} \]
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Time = 9.93 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=-\frac {\frac {2\,A}{b}+\frac {x\,\left (3\,A\,c-B\,b\right )}{b^2}}{b\,\sqrt {x}+c\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (3\,A\,c-B\,b\right )}{b^{5/2}\,\sqrt {c}} \]
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