Integrand size = 29, antiderivative size = 105 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\frac {(A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {a A \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Time = 0.03 (sec) , antiderivative size = 105, 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} \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\frac {x \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {a A \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)} \]
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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} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b (A b+a B)+\frac {a A b}{x}+b^2 B x\right ) \, dx}{a b+b^2 x} \\ & = \frac {(A b+a B) x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {a A \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\frac {\sqrt {(a+b x)^2} (x (2 A b+2 a B+b B x)+2 a A \log (x))}{2 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b^{2} B \,x^{2}+2 A a b \ln \left (-b x \right )+2 A \,b^{2} x +2 B a b x +2 A b a +B \,a^{2}\right )}{2 b}\) | \(54\) |
risch | \(\frac {b B \,x^{2} \sqrt {\left (b x +a \right )^{2}}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A b x}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a B x}{b x +a}+\frac {a A \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(86\) |
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Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.21 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\frac {1}{2} \, B b x^{2} + A a \log \left (x\right ) + {\left (B a + A b\right )} x \]
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\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{x}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B x + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a}{2 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.44 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=\frac {1}{2} \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + B a x \mathrm {sgn}\left (b x + a\right ) + A b x \mathrm {sgn}\left (b x + a\right ) + A a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) \]
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Time = 10.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx=A\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}-A\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )\,\sqrt {a^2}+\frac {B\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b}+\frac {A\,a\,b\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}} \]
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