Integrand size = 29, antiderivative size = 243 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (2 A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (2 A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 243, 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} \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 b \log (x) (a+b x) (2 A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (a+b x) (2 A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (3 A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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^3 \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^3}+\frac {-3 A b+a B}{a^4 b^3 x^2}-\frac {3 (-2 A b+a B)}{a^5 b^2 x}+\frac {-A b+a B}{a^3 b (a+b x)^3}+\frac {-3 A b+2 a B}{a^4 b (a+b x)^2}+\frac {3 (-2 A b+a B)}{a^5 b (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {b (3 A b-2 a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B)}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (2 A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (2 A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-a \left (-12 A b^3 x^3+6 a b^2 x^2 (-3 A+B x)+a^3 (A+2 B x)+a^2 b x (-4 A+9 B x)\right )+6 b (2 A b-a B) x^2 (a+b x)^2 \log (x)+6 b (-2 A b+a B) x^2 (a+b x)^2 \log (a+b x)}{2 a^5 x^2 (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {3 b^{2} \left (2 A b -B a \right ) x^{3}}{a^{4}}+\frac {9 b \left (2 A b -B a \right ) x^{2}}{2 a^{3}}+\frac {\left (2 A b -B a \right ) x}{a^{2}}-\frac {A}{2 a}\right )}{\left (b x +a \right )^{3} x^{2}}-\frac {3 \sqrt {\left (b x +a \right )^{2}}\, b \left (2 A b -B a \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) a^{5}}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, b \left (2 A b -B a \right ) \ln \left (-x \right )}{\left (b x +a \right ) a^{5}}\) | \(153\) |
default | \(\frac {\left (12 A \ln \left (x \right ) x^{4} b^{4}-12 A \ln \left (b x +a \right ) x^{4} b^{4}-6 B \ln \left (x \right ) x^{4} a \,b^{3}+6 B \ln \left (b x +a \right ) x^{4} a \,b^{3}+24 A \ln \left (x \right ) x^{3} a \,b^{3}-24 A \ln \left (b x +a \right ) x^{3} a \,b^{3}-12 B \ln \left (x \right ) x^{3} a^{2} b^{2}+12 B \ln \left (b x +a \right ) x^{3} a^{2} b^{2}+12 A \ln \left (x \right ) x^{2} a^{2} b^{2}-12 A \ln \left (b x +a \right ) x^{2} a^{2} b^{2}+12 a A \,b^{3} x^{3}-6 B \ln \left (x \right ) x^{2} a^{3} b +6 B \ln \left (b x +a \right ) x^{2} a^{3} b -6 x^{3} B \,a^{2} b^{2}+18 a^{2} A \,b^{2} x^{2}-9 x^{2} B \,a^{3} b +4 a^{3} A b x -2 a^{4} B x -A \,a^{4}\right ) \left (b x +a \right )}{2 x^{2} a^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(263\) |
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Time = 0.36 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {A a^{4} + 6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]
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\[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {6 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} - \frac {3 \, B b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}} + \frac {6 \, A b^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}} - \frac {B}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x} + \frac {5 \, A b}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x} - \frac {A}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x^{2}} + \frac {A}{2 \, a^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {B}{2 \, a^{2} b {\left (x + \frac {a}{b}\right )}^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.61 \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, B a b^{2} x^{3} - 12 \, A b^{3} x^{3} + 9 \, B a^{2} b x^{2} - 18 \, A a b^{2} x^{2} + 2 \, B a^{3} x - 4 \, A a^{2} b x + A a^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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