Integrand size = 19, antiderivative size = 86 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=-\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {676, 686, 674, 213} \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\frac {c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {\sqrt {b x+c x^2}}{2 x^{5/2}} \]
[In]
[Out]
Rule 213
Rule 674
Rule 676
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}+\frac {1}{4} c \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {c^2 \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b} \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {c^2 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b} \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}+\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\frac {\sqrt {x (b+c x)} \left (-\sqrt {b} \sqrt {b+c x} (2 b+c x)+c^2 x^2 \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{4 b^{3/2} x^{5/2} \sqrt {b+c x}} \]
[In]
[Out]
Time = 2.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (c x +2 b \right )}{4 x^{\frac {3}{2}} b \sqrt {x \left (c x +b \right )}}+\frac {c^{2} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{4 b^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}\) | \(70\) |
default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{2} x^{2}-c x \sqrt {b}\, \sqrt {c x +b}-2 b^{\frac {3}{2}} \sqrt {c x +b}\right )}{4 b^{\frac {3}{2}} x^{\frac {5}{2}} \sqrt {c x +b}}\) | \(71\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\left [\frac {\sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b^{2} x^{3}}, -\frac {\sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b^{2} x^{3}}\right ] \]
[In]
[Out]
\[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{x^{\frac {7}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x}}{x^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=-\frac {\frac {c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {{\left (c x + b\right )}^{\frac {3}{2}} c^{3} + \sqrt {c x + b} b c^{3}}{b c^{2} x^{2}}}{4 \, c} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{x^{7/2}} \,d x \]
[In]
[Out]