Integrand size = 24, antiderivative size = 97 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \sqrt {x}}{b^2 \sqrt {b x+c x^2}}-\frac {(2 b B-3 A c) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {806, 680, 674, 213} \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {(2 b B-3 A c) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}}+\frac {\sqrt {x} (2 b B-3 A c)}{b^2 \sqrt {b x+c x^2}}-\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}} \]
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Rule 213
Rule 674
Rule 680
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}}+\frac {\left (\frac {1}{2} (b B-2 A c)+\frac {1}{2} (b B-A c)\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{b} \\ & = -\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \sqrt {x}}{b^2 \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{2 b^2} \\ & = -\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \sqrt {x}}{b^2 \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b^2} \\ & = -\frac {A}{b \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-3 A c) \sqrt {x}}{b^2 \sqrt {b x+c x^2}}-\frac {(2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {b} (2 b B x-A (b+3 c x))-(2 b B-3 A c) x \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {x} \sqrt {x (b+c x)}} \]
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Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (3 A \sqrt {c x +b}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c x -2 B \sqrt {c x +b}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) b x +2 B \,b^{\frac {3}{2}} x -3 A \sqrt {b}\, c x -A \,b^{\frac {3}{2}}\right )}{x^{\frac {3}{2}} \left (c x +b \right ) b^{\frac {5}{2}}}\) | \(94\) |
risch | \(-\frac {A \left (c x +b \right )}{b^{2} \sqrt {x}\, \sqrt {x \left (c x +b \right )}}-\frac {\left (-\frac {2 \left (3 A c -2 B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \left (-2 A c +2 B b \right )}{\sqrt {c x +b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{2 b^{2} \sqrt {x \left (c x +b \right )}}\) | \(94\) |
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Time = 0.31 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.56 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{3} + {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (A b^{2} - {\left (2 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{2 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}, \frac {{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{3} + {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (A b^{2} - {\left (2 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{b^{3} c x^{3} + b^{4} x^{2}}\right ] \]
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\[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\sqrt {x} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} \sqrt {x}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {{\left (2 \, B b - 3 \, A c\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {2 \, {\left (c x + b\right )} B b - 2 \, B b^{2} - 3 \, {\left (c x + b\right )} A c + 2 \, A b c}{{\left ({\left (c x + b\right )}^{\frac {3}{2}} - \sqrt {c x + b} b\right )} b^{2}} \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{\sqrt {x}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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