Integrand size = 29, antiderivative size = 108 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 108, 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} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac {b B \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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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^3} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^3}+\frac {b (A b+a B)}{x^2}+\frac {b^2 B}{x}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b 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. \(225\) vs. \(2(108)=216\).
Time = 0.34 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=-\frac {-a \sqrt {a^2} A-2 \sqrt {a^2} A b x-2 a \sqrt {a^2} B x+a A \sqrt {(a+b x)^2}+A b x \sqrt {(a+b x)^2}+2 a B x \sqrt {(a+b x)^2}+4 a b B x^2 \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )+4 \sqrt {a^2} b B x^2 \log (x)-2 \sqrt {a^2} b B x^2 \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )-2 \sqrt {a^2} b B x^2 \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{4 a x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.35
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (-2 B \ln \left (-b x \right ) b \,x^{2}+2 A b x +2 a B x +a A \right )}{2 x^{2}}\) | \(38\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-A b -B a \right ) x -\frac {a A}{2}\right )}{\left (b x +a \right ) x^{2}}+\frac {b B \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(59\) |
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.27 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=\frac {2 \, B b x^{2} \log \left (x\right ) - A a - 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (73) = 146\).
Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} B b \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{2 \, a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B}{x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b}{2 \, a x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{2 \, a^{2} x^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.46 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=B b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {A a \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (B a \mathrm {sgn}\left (b x + a\right ) + A b \mathrm {sgn}\left (b x + a\right )\right )} x}{2 \, x^{2}} \]
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Time = 10.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx=B\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )\,\sqrt {b^2}-\frac {B\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}-\frac {A\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+2\,b\,x\right )}{2\,x^2\,\left (a+b\,x\right )}-\frac {B\,a\,b\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \]
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