Integrand size = 27, antiderivative size = 113 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {A b-2 a B}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {784, 78} \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {A b-2 a B}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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 {x (A+B 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 b+a B)}{b^5 (a+b x)^3}+\frac {A b-2 a B}{b^5 (a+b x)^2}+\frac {B}{b^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {A b-2 a B}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.65 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {\frac {b x \left (2 a^5 B-a^3 A b^2 x+3 a^4 b B x-a A b^4 x^3+a^2 b^3 B x^3-\sqrt {a^2} \sqrt {(a+b x)^2} \left (2 a^3 B+a^2 b B x+A b^3 x^2-a b^2 x (A+B x)\right )\right )}{a^3 (a+b x) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}-4 B \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )}{2 b^3} \]
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Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {\left (A b -2 B a \right ) x}{b^{2}}-\frac {a \left (A b -3 B a \right )}{2 b^{3}}\right )}{\left (b x +a \right )^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, B \ln \left (b x +a \right )}{\left (b x +a \right ) b^{3}}\) | \(75\) |
default | \(-\frac {\left (-2 B \ln \left (b x +a \right ) b^{2} x^{2}-4 B \ln \left (b x +a \right ) x a b +2 A \,b^{2} x -2 B \ln \left (b x +a \right ) a^{2}-4 B a b x +A b a -3 B \,a^{2}\right ) \left (b x +a \right )}{2 b^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(83\) |
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Time = 0.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 \, B a^{2} - A a b + 2 \, {\left (2 \, B a b - A b^{2}\right )} x + 2 \, {\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
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\[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {x \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {B \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {2 \, B a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, B a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {A a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {B \log \left ({\left | b x + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (2 \, B a - A b\right )} x + \frac {3 \, B a^{2} - A a b}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {x\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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