Integrand size = 29, antiderivative size = 196 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(3 A b-a B) (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 196, 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} \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\log (x) (a+b x) (3 A b-a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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^2 \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^2}+\frac {-3 A b+a B}{a^4 b^3 x}+\frac {A b-a B}{a^2 b^2 (a+b x)^3}+\frac {2 A b-a B}{a^3 b^2 (a+b x)^2}+\frac {3 A b-a B}{a^4 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 A b-a B}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(3 A b-a B) (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {a \left (-6 A b^2 x^2+a b x (-9 A+2 B x)+a^2 (-2 A+3 B x)\right )+2 (-3 A b+a B) x (a+b x)^2 \log (x)+2 (3 A b-a B) x (a+b x)^2 \log (a+b x)}{2 a^4 x (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 0.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b \left (3 A b -B a \right ) x^{2}}{a^{3}}-\frac {3 \left (3 A b -B a \right ) x}{2 a^{2}}-\frac {A}{a}\right )}{\left (b x +a \right )^{3} x}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A b -B a \right ) \ln \left (-b x -a \right )}{\left (b x +a \right ) a^{4}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 A b -B a \right ) \ln \left (x \right )}{\left (b x +a \right ) a^{4}}\) | \(132\) |
default | \(-\frac {\left (6 A \ln \left (x \right ) x^{3} b^{3}-6 A \ln \left (b x +a \right ) x^{3} b^{3}-2 B \ln \left (x \right ) x^{3} a \,b^{2}+2 B \ln \left (b x +a \right ) x^{3} a \,b^{2}+12 A \ln \left (x \right ) x^{2} a \,b^{2}-12 A \ln \left (b x +a \right ) x^{2} a \,b^{2}-4 B \ln \left (x \right ) x^{2} a^{2} b +4 B \ln \left (b x +a \right ) x^{2} a^{2} b +6 A \ln \left (x \right ) x \,a^{2} b -6 A \ln \left (b x +a \right ) x \,a^{2} b +6 A a \,b^{2} x^{2}-2 B \,a^{3} \ln \left (x \right ) x +2 B \ln \left (b x +a \right ) a^{3} x -2 B \,a^{2} b \,x^{2}+9 A \,a^{2} b x -3 a^{3} B x +2 A \,a^{3}\right ) \left (b x +a \right )}{2 x \,a^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(221\) |
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Time = 0.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]
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\[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} + \frac {B}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}} - \frac {3 \, A b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}} - \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x} + \frac {B}{2 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {A}{2 \, a^{2} b {\left (x + \frac {a}{b}\right )}^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x}{2 \, {\left (b x + a\right )}^{2} a^{4} x \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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