Integrand size = 24, antiderivative size = 71 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]
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Time = 0.01 (sec) , antiderivative size = 71, 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} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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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 {a b+b^2 x}{x^4} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {\sqrt {(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.28
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (3 b x +2 a \right )}{6 x^{3}}\) | \(20\) |
risch | \(\frac {\left (-\frac {b x}{2}-\frac {a}{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{3} \left (b x +a \right )}\) | \(29\) |
gosper | \(-\frac {\left (3 b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 x^{3} \left (b x +a \right )}\) | \(30\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \]
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\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{2}}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (45) = 90\).
Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}}{2 \, a^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{2 \, a^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=\frac {b^{3} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {3 \, b x \mathrm {sgn}\left (b x + a\right ) + 2 \, a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \]
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Time = 9.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx=-\frac {\left (2\,a+3\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{6\,x^3\,\left (a+b\,x\right )} \]
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