Integrand size = 17, antiderivative size = 48 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}+\frac {4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=\frac {4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}-\frac {(2 c) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{5 b} \\ & = -\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}+\frac {4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (3 b^2+b c x-2 c^2 x^2\right )}{15 b^2 x^3} \]
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Time = 2.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(-\frac {2 \left (c x +b \right ) \sqrt {x \left (c x +b \right )}\, \left (-\frac {2 c x}{3}+b \right )}{5 x^{3} b^{2}}\) | \(29\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +3 b \right ) \sqrt {c \,x^{2}+b x}}{15 b^{2} x^{3}}\) | \(33\) |
trager | \(-\frac {2 \left (-2 c^{2} x^{2}+b c x +3 b^{2}\right ) \sqrt {c \,x^{2}+b x}}{15 b^{2} x^{3}}\) | \(38\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\) | \(41\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-2 c^{2} x^{2}+b c x +3 b^{2}\right )}{15 x^{2} \sqrt {x \left (c x +b \right )}\, b^{2}}\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=\frac {2 \, {\left (2 \, c^{2} x^{2} - b c x - 3 \, b^{2}\right )} \sqrt {c x^{2} + b x}}{15 \, b^{2} x^{3}} \]
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\[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{x^{4}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=\frac {4 \, \sqrt {c x^{2} + b x} c^{2}}{15 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x} c}{15 \, b x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x}}{5 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (40) = 80\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=\frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{\frac {3}{2}} + 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} \sqrt {c} + 3 \, b^{3}\right )}}{15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5}} \]
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Time = 9.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx=-\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (3\,b^2+b\,c\,x-2\,c^2\,x^2\right )}{15\,b^2\,x^3} \]
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