Integrand size = 17, antiderivative size = 48 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5} \]
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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} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}-\frac {(2 c) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{7 b} \\ & = -\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=-\frac {2 (5 b-2 c x) (x (b+c x))^{5/2}}{35 b^2 x^6} \]
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Time = 2.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {2 \left (c x +b \right )^{2} \left (-\frac {2 c x}{5}+b \right ) \sqrt {x \left (c x +b \right )}}{7 x^{4} b^{2}}\) | \(31\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +5 b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{35 b^{2} x^{5}}\) | \(33\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\) | \(41\) |
trager | \(-\frac {2 \left (-2 c^{3} x^{3}+b \,c^{2} x^{2}+8 b^{2} c x +5 b^{3}\right ) \sqrt {c \,x^{2}+b x}}{35 b^{2} x^{4}}\) | \(49\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-2 c^{3} x^{3}+b \,c^{2} x^{2}+8 b^{2} c x +5 b^{3}\right )}{35 x^{3} \sqrt {x \left (c x +b \right )}\, b^{2}}\) | \(52\) |
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none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (2 \, c^{3} x^{3} - b c^{2} x^{2} - 8 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{35 \, b^{2} x^{4}} \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (40) = 80\).
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {4 \, \sqrt {c x^{2} + b x} c^{3}}{35 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x} c^{2}}{35 \, b x^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} c}{70 \, x^{3}} + \frac {3 \, \sqrt {c x^{2} + b x} b}{14 \, x^{4}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{2 \, x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (40) = 80\).
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.44 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b c^{2} + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} c^{\frac {3}{2}} + 98 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{3} c + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{4} \sqrt {c} + 5 \, b^{5}\right )}}{35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7}} \]
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Time = 9.53 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.65 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx=\frac {4\,c^3\,\sqrt {c\,x^2+b\,x}}{35\,b^2\,x}-\frac {16\,c\,\sqrt {c\,x^2+b\,x}}{35\,x^3}-\frac {2\,c^2\,\sqrt {c\,x^2+b\,x}}{35\,b\,x^2}-\frac {2\,b\,\sqrt {c\,x^2+b\,x}}{7\,x^4} \]
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