Integrand size = 17, antiderivative size = 72 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\frac {3}{4} b \sqrt {b x+c x^2}+\frac {\left (b x+c x^2\right )^{3/2}}{2 x}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {678, 634, 212} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}}+\frac {3}{4} b \sqrt {b x+c x^2}+\frac {\left (b x+c x^2\right )^{3/2}}{2 x} \]
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Rule 212
Rule 634
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x+c x^2\right )^{3/2}}{2 x}+\frac {1}{4} (3 b) \int \frac {\sqrt {b x+c x^2}}{x} \, dx \\ & = \frac {3}{4} b \sqrt {b x+c x^2}+\frac {\left (b x+c x^2\right )^{3/2}}{2 x}+\frac {1}{8} \left (3 b^2\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx \\ & = \frac {3}{4} b \sqrt {b x+c x^2}+\frac {\left (b x+c x^2\right )^{3/2}}{2 x}+\frac {1}{4} \left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right ) \\ & = \frac {3}{4} b \sqrt {b x+c x^2}+\frac {\left (b x+c x^2\right )^{3/2}}{2 x}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{4} \sqrt {x (b+c x)} \left (5 b+2 c x-\frac {3 b^2 \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {c} \sqrt {x} \sqrt {b+c x}}\right ) \]
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Time = 2.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {\left (2 c x +5 b \right ) x \left (c x +b \right )}{4 \sqrt {x \left (c x +b \right )}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 \sqrt {c}}\) | \(59\) |
pseudoelliptic | \(\frac {2 c^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}\, x +3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) b^{2}+5 b \sqrt {c}\, \sqrt {x \left (c x +b \right )}}{4 \sqrt {c}}\) | \(59\) |
default | \(\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{b \,x^{2}}-\frac {6 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2}\right )}{b}\) | \(99\) |
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Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.75 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\left [\frac {3 \, b^{2} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, c^{2} x + 5 \, b c\right )} \sqrt {c x^{2} + b x}}{8 \, c}, -\frac {3 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, c^{2} x + 5 \, b c\right )} \sqrt {c x^{2} + b x}}{4 \, c}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\frac {3 \, b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, \sqrt {c}} + \frac {3}{4} \, \sqrt {c x^{2} + b x} b + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{2 \, x} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=-\frac {3 \, b^{2} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{8 \, \sqrt {c}} + \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (2 \, c x + 5 \, b\right )} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^2} \,d x \]
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