Integrand size = 17, antiderivative size = 100 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7} \]
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Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}-\frac {(6 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{13 a} \\ & = -\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}+\frac {\left (24 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{143 a^2} \\ & = -\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}-\frac {\left (16 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{429 a^3} \\ & = -\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {2 (x (a+b x))^{7/2} \left (231 a^3-126 a^2 b x+56 a b^2 x^2-16 b^3 x^3\right )}{3003 a^4 x^{10}} \]
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Time = 2.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{3003 a^{4} x^{9}}\) | \(55\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{3} \sqrt {x \left (b x +a \right )}\, \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right )}{3003 x^{7} a^{4}}\) | \(55\) |
trager | \(-\frac {2 \left (-16 b^{6} x^{6}+8 a \,x^{5} b^{5}-6 a^{2} x^{4} b^{4}+5 a^{3} x^{3} b^{3}+371 a^{4} x^{2} b^{2}+567 a^{5} x b +231 a^{6}\right ) \sqrt {b \,x^{2}+a x}}{3003 a^{4} x^{7}}\) | \(83\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{6} x^{6}+8 a \,x^{5} b^{5}-6 a^{2} x^{4} b^{4}+5 a^{3} x^{3} b^{3}+371 a^{4} x^{2} b^{2}+567 a^{5} x b +231 a^{6}\right )}{3003 x^{6} \sqrt {x \left (b x +a \right )}\, a^{4}}\) | \(86\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{13 a \,x^{10}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{11 a \,x^{9}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{9 a \,x^{8}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {2 \, {\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt {b x^{2} + a x}}{3003 \, a^{4} x^{7}} \]
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\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{10}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (84) = 168\).
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} b^{6}}{3003 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{3003 \, a^{3} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{4}}{1001 \, a^{2} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} b^{3}}{3003 \, a x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{1716 \, x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{1144 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{104 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{4 \, x^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.81 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {2 \, {\left (6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} b^{\frac {9}{2}} + 36036 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{2} b^{\frac {7}{2}} + 161733 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{4} b^{\frac {5}{2}} + 121121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{6} b^{\frac {3}{2}} + 17199 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{8} \sqrt {b} + 231 \, a^{9}\right )}}{3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{13}} \]
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Time = 10.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {4\,b^4\,\sqrt {b\,x^2+a\,x}}{1001\,a^2\,x^3}-\frac {106\,b^2\,\sqrt {b\,x^2+a\,x}}{429\,x^5}-\frac {10\,b^3\,\sqrt {b\,x^2+a\,x}}{3003\,a\,x^4}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{3003\,a^3\,x^2}+\frac {32\,b^6\,\sqrt {b\,x^2+a\,x}}{3003\,a^4\,x}-\frac {54\,a\,b\,\sqrt {b\,x^2+a\,x}}{143\,x^6} \]
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