Integrand size = 24, antiderivative size = 223 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {b^5 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^6} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 b^5}{x^6}+\frac {5 a^4 b^6}{x^5}+\frac {10 a^3 b^7}{x^4}+\frac {10 a^2 b^8}{x^3}+\frac {5 a b^9}{x^2}+\frac {b^{10}}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {1}{120} \left (-\frac {\sqrt {(a+b x)^2} \left (12 a^4+63 a^3 b x+137 a^2 b^2 x^2+163 a b^3 x^3+137 b^4 x^4\right )}{x^5}+\frac {\sqrt {a^2} \left (12 a^4+75 a^3 b x+200 a^2 b^2 x^2+300 a b^3 x^3+300 b^4 x^4\right )}{x^5}-120 b^5 \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-\frac {120 \sqrt {a^2} b^5 \log (x)}{a}+\frac {60 \sqrt {a^2} b^5 \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )}{a}+\frac {60 \sqrt {a^2} b^5 \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{a}\right ) \]
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Time = 2.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.34
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 b^{5} \ln \left (x \right ) x^{5}-300 a \,b^{4} x^{4}-300 a^{2} b^{3} x^{3}-200 a^{3} b^{2} x^{2}-75 a^{4} b x -12 a^{5}\right )}{60 x^{5} \left (b x +a \right )^{5}}\) | \(76\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{5} a^{5}-5 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}-\frac {10}{3} a^{3} b^{2} x^{2}-\frac {5}{4} a^{4} b x \right )}{\left (b x +a \right ) x^{5}}+\frac {b^{5} \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(88\) |
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.26 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {60 \, b^{5} x^{5} \log \left (x\right ) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (151) = 302\).
Time = 0.21 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} b^{5} \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{5} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{6} x}{2 \, a^{2}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}}{2 \, a} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{6} x}{4 \, a^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5}}{12 \, a^{3}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{5}}{15 \, a^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{3 \, a^{4} x} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{15 \, a^{5} x^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{60 \, a^{4} x^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{20 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{5 \, a^{2} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=b^{5} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {300 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 300 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 200 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 75 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{60 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^6} \,d x \]
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