Integrand size = 17, antiderivative size = 68 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=-\frac {2 c \sqrt {b x+c x^2}}{x}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 68, 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 = {676, 634, 212} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 c \sqrt {b x+c x^2}}{x}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rule 212
Rule 634
Rule 676
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+c \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx \\ & = -\frac {2 c \sqrt {b x+c x^2}}{x}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+c^2 \int \frac {1}{\sqrt {b x+c x^2}} \, dx \\ & = -\frac {2 c \sqrt {b x+c x^2}}{x}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right ) \\ & = -\frac {2 c \sqrt {b x+c x^2}}{x}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.16 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (\sqrt {b+c x} (b+4 c x)+3 c^{3/2} x^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{3 x^2 \sqrt {b+c x}} \]
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Time = 2.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(\frac {6 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) x^{2}-2 \sqrt {x \left (c x +b \right )}\, \left (4 c x +b \right )}{3 x^{2}}\) | \(49\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (4 c x +b \right )}{3 x \sqrt {x \left (c x +b \right )}}+c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )\) | \(55\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{3 b \,x^{4}}+\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{b \,x^{3}}+\frac {4 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{b \,x^{2}}-\frac {6 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2}\right )}{b}\right )}{b}\right )}{3 b}\) | \(151\) |
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Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.71 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=\left [\frac {3 \, c^{\frac {3}{2}} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x} {\left (4 \, c x + b\right )}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-c} c x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x} {\left (4 \, c x + b\right )}\right )}}{3 \, x^{2}}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=c^{\frac {3}{2}} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {7 \, \sqrt {c x^{2} + b x} c}{3 \, x} - \frac {\sqrt {c x^{2} + b x} b}{3 \, x^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.69 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=-c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, {\left (6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} \sqrt {c} + b^{3}\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^4} \,d x \]
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