Integrand size = 29, antiderivative size = 113 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x}-\frac {(A b-a B) (a+b x) \log (x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {783, 660, 36, 29, 31} \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {\log (x) (a+b x) (A b-a B)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x} \]
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Rule 29
Rule 31
Rule 36
Rule 660
Rule 783
Rubi steps \begin{align*} \text {integral}& = -\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {1}{x \sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{2 a b} \\ & = -\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x}-\frac {\left (\left (2 A b^2-2 a b B\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x \left (a b+b^2 x\right )} \, dx}{2 a b \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x}+\frac {\left (\left (2 A b^2-2 a b B\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{a b+b^2 x} \, dx}{2 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (\left (2 A b^2-2 a b B\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{x} \, dx}{2 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {A \sqrt {a^2+2 a b x+b^2 x^2}}{a^2 x}-\frac {(A b-a B) (a+b x) \log (x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {a^2 A-\sqrt {a^2} A \sqrt {(a+b x)^2}+2 a (A b-a B) x \log (x)-\left (-a+\sqrt {a^2}\right ) (-A b+a B) x \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+\left (a+\sqrt {a^2}\right ) (-A b+a B) x \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{2 \left (a^2\right )^{3/2} x} \]
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Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (A \ln \left (x \right ) x b -A \ln \left (b x +a \right ) x b -B a \ln \left (x \right ) x +B \ln \left (b x +a \right ) a x +a A \right )}{\sqrt {\left (b x +a \right )^{2}}\, x \,a^{2}}\) | \(61\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, A}{\left (b x +a \right ) a x}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A b -B a \right ) \ln \left (-b x -a \right )}{\left (b x +a \right ) a^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A b -B a \right ) \ln \left (x \right )}{\left (b x +a \right ) a^{2}}\) | \(95\) |
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Time = 0.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.36 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {{\left (B a - A b\right )} x \log \left (b x + a\right ) - {\left (B a - A b\right )} x \log \left (x\right ) + A a}{a^{2} x} \]
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\[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A + B x}{x^{2} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a} + \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{a^{2} x} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {{\left (B a \mathrm {sgn}\left (b x + a\right ) - A b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {A \mathrm {sgn}\left (b x + a\right )}{a x} - \frac {{\left (B a b \mathrm {sgn}\left (b x + a\right ) - A b^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b} \]
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Time = 10.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {A\,a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a}{\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}\right )}{{\left (a^2\right )}^{3/2}}-\frac {A\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{a^2\,x}-\frac {B\,\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \]
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