Integrand size = 24, antiderivative size = 194 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 194, 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, 46} \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\log (x) (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{x \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{a^5 b^5 x}-\frac {1}{a b^4 (a+b x)^5}-\frac {1}{a^2 b^4 (a+b x)^4}-\frac {1}{a^3 b^4 (a+b x)^3}-\frac {1}{a^4 b^4 (a+b x)^2}-\frac {1}{a^5 b^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {a \left (25 a^3+52 a^2 b x+42 a b^2 x^2+12 b^3 x^3\right )+12 (a+b x)^4 \log (x)-12 (a+b x)^4 \log (a+b x)}{12 a^5 (a+b x)^3 \sqrt {(a+b x)^2}} \]
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Time = 2.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b^{3} x^{3}}{a^{4}}+\frac {7 b^{2} x^{2}}{2 a^{3}}+\frac {13 b x}{3 a^{2}}+\frac {25}{12 a}\right )}{\left (b x +a \right )^{5}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (b x +a \right )}{\left (b x +a \right ) a^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (-x \right )}{\left (b x +a \right ) a^{5}}\) | \(104\) |
default | \(\frac {\left (12 \ln \left (x \right ) b^{4} x^{4}-12 \ln \left (b x +a \right ) b^{4} x^{4}+48 \ln \left (x \right ) a \,b^{3} x^{3}-48 \ln \left (b x +a \right ) a \,b^{3} x^{3}+72 \ln \left (x \right ) a^{2} b^{2} x^{2}-72 \ln \left (b x +a \right ) a^{2} b^{2} x^{2}+12 a \,b^{3} x^{3}+48 \ln \left (x \right ) a^{3} b x -48 \ln \left (b x +a \right ) a^{3} b x +42 a^{2} b^{2} x^{2}+12 \ln \left (x \right ) a^{4}-12 a^{4} \ln \left (b x +a \right )+52 a^{3} b x +25 a^{4}\right ) \left (b x +a \right )}{12 a^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(173\) |
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Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4} - 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\right )}} \]
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\[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} + \frac {1}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}} + \frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}} + \frac {1}{2 \, a^{3} b^{2} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {1}{4 \, a b^{4} {\left (x + \frac {a}{b}\right )}^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {\log \left ({\left | x \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4}}{12 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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