Integrand size = 29, antiderivative size = 140 \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 140, 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, 78} \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x \left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {A}{a^3 b^3 x}+\frac {-A b+a B}{a b^3 (a+b x)^3}-\frac {A}{a^2 b^2 (a+b x)^2}-\frac {A}{a^3 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {A}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {a \left (3 a A b-a^2 B+2 A b^2 x\right )+2 A b (a+b x)^2 \log (x)-2 A b (a+b x)^2 \log (a+b x)}{2 a^3 b (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {A b x}{a^{2}}+\frac {3 A b -B a}{2 b a}\right )}{\left (b x +a \right )^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \ln \left (-x \right )}{\left (b x +a \right ) a^{3}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, A \ln \left (b x +a \right )}{\left (b x +a \right ) a^{3}}\) | \(97\) |
default | \(\frac {\left (2 A \ln \left (x \right ) x^{2} b^{3}-2 A \ln \left (b x +a \right ) x^{2} b^{3}+4 A \ln \left (x \right ) x a \,b^{2}-4 A \ln \left (b x +a \right ) x a \,b^{2}+2 A \ln \left (x \right ) a^{2} b -2 A \ln \left (b x +a \right ) a^{2} b +2 A x a \,b^{2}+3 A \,a^{2} b -B \,a^{3}\right ) \left (b x +a \right )}{2 b \,a^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(117\) |
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Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \]
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\[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}} - \frac {B}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {A}{2 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59 \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {A \log \left ({\left | b x + a \right |}\right )}{a^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {A \log \left ({\left | x \right |}\right )}{a^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b}{2 \, {\left (b x + a\right )}^{2} a^{3} b \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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