Integrand size = 28, antiderivative size = 131 \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {a+b x}{(b d-a e) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) \log (d+e x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 46} \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {a+b x}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac {b (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {b (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^2} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (a b+b^2 x\right ) \int \left (\frac {b}{(b d-a e)^2 (a+b x)}-\frac {e}{b (b d-a e) (d+e x)^2}-\frac {e}{(b d-a e)^2 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {a+b x}{(b d-a e) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) \log (d+e x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.89 \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {-\frac {\sqrt {a^2} e (-b d+a e) x}{a d (d+e x)}-\frac {e \sqrt {(a+b x)^2}}{d+e x}+\frac {\left (a-\sqrt {a^2}\right ) b \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )}{a}-\frac {\left (a+\sqrt {a^2}\right ) b \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{a}+\frac {\left (a+\sqrt {a^2}\right ) b \log \left (2 a e x+d \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )}{a}+\frac {\left (-a+\sqrt {a^2}\right ) b \log \left (-2 a e x+d \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{a}}{2 (b d-a e)^2} \]
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Time = 2.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (\ln \left (b x +a \right ) b e x -\ln \left (e x +d \right ) b e x +\ln \left (b x +a \right ) b d -\ln \left (e x +d \right ) b d -a e +b d \right )}{\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{2} \left (e x +d \right )}\) | \(81\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{\left (b x +a \right ) \left (a e -b d \right ) \left (e x +d \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(135\) |
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Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {b d - a e + {\left (b e x + b d\right )} \log \left (b x + a\right ) - {\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} + {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \]
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\[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx={\left (\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} - \frac {b e \log \left ({\left | e x + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac {1}{{\left (b d - a e\right )} {\left (e x + d\right )}}\right )} \mathrm {sgn}\left (b x + a\right ) \]
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Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^2} \,d x \]
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