Integrand size = 19, antiderivative size = 56 \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=-\frac {\sqrt {b x+c x^2}}{b x^{3/2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {686, 674, 213} \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {c \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {\sqrt {b x+c x^2}}{b x^{3/2}} \]
[In]
[Out]
Rule 213
Rule 674
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b x+c x^2}}{b x^{3/2}}-\frac {c \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{2 b} \\ & = -\frac {\sqrt {b x+c x^2}}{b x^{3/2}}-\frac {c \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b} \\ & = -\frac {\sqrt {b x+c x^2}}{b x^{3/2}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {-\sqrt {b} (b+c x)+c x \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {x} \sqrt {x (b+c x)}} \]
[In]
[Out]
Time = 2.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c x -\sqrt {c x +b}\, \sqrt {b}\right )}{b^{\frac {3}{2}} x^{\frac {3}{2}} \sqrt {c x +b}}\) | \(52\) |
risch | \(-\frac {c x +b}{b \sqrt {x}\, \sqrt {x \left (c x +b \right )}}+\frac {c \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{b^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}\) | \(60\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.25 \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\left [\frac {\sqrt {b} c x^{2} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, \sqrt {c x^{2} + b x} b \sqrt {x}}{2 \, b^{2} x^{2}}, -\frac {\sqrt {-b} c x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} b \sqrt {x}}{b^{2} x^{2}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \sqrt {x \left (b + c x\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} x^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=-\frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {\sqrt {c x + b} c}{b x}}{c} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{3/2}\,\sqrt {c\,x^2+b\,x}} \,d x \]
[In]
[Out]