Integrand size = 29, antiderivative size = 103 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {784, 77} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=\frac {\log (x) \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rule 77
Rule 784
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 ) (A+B x)}{x^2} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^2 B+\frac {a A b}{x^2}+\frac {b (A b+a B)}{x}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(761\) vs. \(2(103)=206\).
Time = 1.21 (sec) , antiderivative size = 761, normalized size of antiderivative = 7.39 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=\frac {2 \left (a^2\right )^{3/2} A+3 a \sqrt {a^2} A b x+\sqrt {a^2} A b^2 x^2-2 a \sqrt {a^2} b B x^2-2 \sqrt {a^2} b^2 B x^3-2 a^2 A \sqrt {(a+b x)^2}-a A b x \sqrt {(a+b x)^2}+2 a b B x^2 \sqrt {(a+b x)^2}-2 (A b+a B) x \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right ) \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-2 (A b+a B) x \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right ) \log (x)+\left (a^2\right )^{3/2} B x \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+a \sqrt {a^2} b B x^2 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )-a^2 B x \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+a \sqrt {a^2} A b x \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )+\sqrt {a^2} A b^2 x^2 \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )-a A b x \sqrt {(a+b x)^2} \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )+\left (a^2\right )^{3/2} B x \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )+a \sqrt {a^2} b B x^2 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )-a^2 B x \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )+a \sqrt {a^2} A b x \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )+\sqrt {a^2} A b^2 x^2 \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )-a A b x \sqrt {(a+b x)^2} \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{2 x \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.43
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (A \ln \left (-b x \right ) b x +B \ln \left (-b x \right ) a x +B b \,x^{2}+a B x -a A \right )}{x}\) | \(44\) |
risch | \(-\frac {a A \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {b B x \sqrt {\left (b x +a \right )^{2}}}{b x +a}+\frac {\left (A b +B a \right ) \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(71\) |
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=\frac {B b x^{2} + {\left (B a + A b\right )} x \log \left (x\right ) - A a}{x} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (70) = 140\).
Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a \log \left (2 \, b^{2} x + 2 \, a b\right ) + \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A b \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{x} \]
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.46 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=B b x \mathrm {sgn}\left (b x + a\right ) + {\left (B a \mathrm {sgn}\left (b x + a\right ) + A b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac {A a \mathrm {sgn}\left (b x + a\right )}{x} \]
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Time = 10.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^2} \, dx=B\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}-B\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )\,\sqrt {a^2}+A\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )\,\sqrt {b^2}-\frac {A\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}+\frac {B\,a\,b\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}}-\frac {A\,a\,b\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \]
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