Integrand size = 24, antiderivative size = 142 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.02 (sec) , antiderivative size = 142, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]
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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 )^3}{x^2} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (3 a b^5+\frac {a^3 b^3}{x^2}+\frac {3 a^2 b^4}{x}+b^6 x\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 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 = 56, normalized size of antiderivative = 0.39 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (-2 a^3+6 a b^2 x^2+b^3 x^3+6 a^2 b x \log (x)\right )}{2 x (a+b x)} \]
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Time = 2.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.37
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{3}+6 a^{2} b \ln \left (x \right ) x +6 a \,b^{2} x^{2}-2 a^{3}\right )}{2 x \left (b x +a \right )^{3}}\) | \(53\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (\frac {1}{2} b \,x^{2}+3 a x \right )}{b x +a}-\frac {a^{3} \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {3 a^{2} b \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(81\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.25 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \left (x\right ) - 2 \, a^{3}}{2 \, x} \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{2} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{2} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2} x + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a b - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} x \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x + a\right )}{x} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^2} \,d x \]
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