Integrand size = 24, antiderivative size = 220 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 220, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\frac {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {5 a^4 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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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^2} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (10 a^3 b^7+\frac {a^5 b^5}{x^2}+\frac {5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\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}}{x (a+b x)}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.36 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (-12 a^5+120 a^3 b^2 x^2+60 a^2 b^3 x^3+20 a b^4 x^4+3 b^5 x^5+60 a^4 b x \log (x)\right )}{12 x (a+b x)} \]
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Time = 2.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (3 b^{5} x^{5}+20 a \,b^{4} x^{4}+60 a^{2} b^{3} x^{3}+60 a^{4} b \ln \left (x \right ) x +120 a^{3} b^{2} x^{2}-12 a^{5}\right )}{12 x \left (b x +a \right )^{5}}\) | \(76\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (\frac {1}{4} b^{3} x^{4}+\frac {5}{3} a \,b^{2} x^{3}+5 a^{2} b \,x^{2}+10 a^{3} x \right )}{b x +a}-\frac {a^{5} \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {5 a^{4} b \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(103\) |
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\frac {3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{4} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{4} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} b^{2} x + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} b + \frac {5}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2} x + \frac {35}{12} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a b - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\frac {1}{4} \, b^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} x \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {a^{5} \mathrm {sgn}\left (b x + a\right )}{x} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^2} \,d x \]
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