Integrand size = 19, antiderivative size = 76 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}-2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 76, 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 = {678, 674, 213} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=-2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \]
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Rule 213
Rule 674
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx \\ & = \frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx \\ & = \frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right ) \\ & = \frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}-2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2 \sqrt {x} \sqrt {b+c x} \left (\sqrt {b+c x} (4 b+c x)-3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{3 \sqrt {x (b+c x)}} \]
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Time = 2.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (3 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-c x \sqrt {c x +b}-4 b \sqrt {c x +b}\right )}{3 \sqrt {x}\, \sqrt {c x +b}}\) | \(61\) |
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Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.70 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\left [\frac {3 \, b^{\frac {3}{2}} x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + b x} {\left (c x + 4 \, b\right )} \sqrt {x}}{3 \, x}, \frac {2 \, {\left (3 \, \sqrt {-b} b x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} {\left (c x + 4 \, b\right )} \sqrt {x}\right )}}{3 \, x}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{\frac {5}{2}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + \frac {2}{3} \, {\left (c x + b\right )}^{\frac {3}{2}} + 2 \, \sqrt {c x + b} b - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )}}{3 \, \sqrt {-b}} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^{5/2}} \,d x \]
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