Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {686, 674, 213} \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}} \]
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Rule 213
Rule 674
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}-\frac {(3 c) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{4 b} \\ & = -\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}+\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^2} \\ & = -\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^2} \\ & = -\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {b} \left (-2 b^2+b c x+3 c^2 x^2\right )-3 c^2 x^2 \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{4 b^{5/2} x^{3/2} \sqrt {x (b+c x)}} \]
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Time = 2.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (-3 c x +2 b \right )}{4 b^{2} x^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}-\frac {3 c^{2} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{4 b^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}}\) | \(71\) |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{2} x^{2}-3 c x \sqrt {b}\, \sqrt {c x +b}+2 b^{\frac {3}{2}} \sqrt {c x +b}\right )}{4 b^{\frac {5}{2}} x^{\frac {5}{2}} \sqrt {c x +b}}\) | \(72\) |
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Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\left [\frac {3 \, \sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (3 \, b c x - 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b^{3} x^{3}}, \frac {3 \, \sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, b c x - 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b^{3} x^{3}}\right ] \]
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\[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )}}\, dx \]
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\[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} x^{\frac {5}{2}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{3} - 5 \, \sqrt {c x + b} b c^{3}}{b^{2} c^{2} x^{2}}}{4 \, c} \]
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Timed out. \[ \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{5/2}\,\sqrt {c\,x^2+b\,x}} \,d x \]
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