Integrand size = 17, antiderivative size = 23 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {664} \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]
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Rule 664
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2 (x (a+b x))^{7/2}}{7 a x^7} \]
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Time = 2.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{7 a \,x^{7}}\) | \(20\) |
gosper | \(-\frac {2 \left (b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 x^{6} a}\) | \(25\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{3} \sqrt {x \left (b x +a \right )}}{7 x^{4} a}\) | \(25\) |
trager | \(-\frac {2 \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {b \,x^{2}+a x}}{7 a \,x^{4}}\) | \(47\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )}{7 x^{3} \sqrt {x \left (b x +a \right )}\, a}\) | \(50\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b x^{2} + a x}}{7 \, a x^{4}} \]
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\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.87 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{7 \, a x} + \frac {\sqrt {b x^{2} + a x} b^{2}}{7 \, x^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{28 \, x^{3}} - \frac {15 \, \sqrt {b x^{2} + a x} a^{2}}{28 \, x^{4}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{4 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 8.35 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=\frac {2 \, {\left (7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{3} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {5}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} b^{\frac {3}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} b + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{5} \sqrt {b} + a^{6}\right )}}{7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7}} \]
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Time = 9.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.43 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {6\,b^2\,\sqrt {b\,x^2+a\,x}}{7\,x^2}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{7\,a\,x}-\frac {6\,a\,b\,\sqrt {b\,x^2+a\,x}}{7\,x^3} \]
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