Integrand size = 30, antiderivative size = 33 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=\frac {2 \text {arctanh}\left (\frac {-\sqrt {b^2-4 a b^3}+2 b^2 x}{b}\right )}{b} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {630, 31} \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=\frac {\log \left (-\sqrt {b^2-4 a b^3}+2 b^2 x+b\right )}{b}-\frac {\log \left (\sqrt {b^2-4 a b^3}-2 b^2 x+b\right )}{b} \]
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Rule 31
Rule 630
Rubi steps \begin{align*} \text {integral}& = -\left (b \int \frac {1}{\frac {1}{2} \left (-b+\sqrt {b^2-4 a b^3}\right )-b^2 x} \, dx\right )+b \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a b^3}\right )-b^2 x} \, dx \\ & = -\frac {\log \left (b+\sqrt {b^2-4 a b^3}-2 b^2 x\right )}{b}+\frac {\log \left (b-\sqrt {b^2-4 a b^3}+2 b^2 x\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(120\) vs. \(2(33)=66\).
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.64 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=\frac {-2 \sqrt {b^2-4 a b^3} \arctan \left (\frac {-1+2 b x}{\sqrt {-1+4 a b}}\right )+2 \sqrt {b^2-4 a b^3} \arctan \left (\frac {1+2 b x}{\sqrt {-1+4 a b}}\right )+b \sqrt {-1+4 a b} \left (\log \left (a+x+b x^2\right )-\log (a+x (-1+b x))\right )}{2 b^2 \sqrt {-1+4 a b}} \]
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Time = 2.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {-2 b^{2} x +\sqrt {-b^{2} \left (4 a b -1\right )}}{b}\right )}{b}\) | \(31\) |
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (31) = 62\).
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=\frac {\log \left (\frac {2 \, b^{2} x + b - \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right ) - \log \left (\frac {2 \, b^{2} x - b - \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=- \frac {\log {\left (x - \frac {1}{2 b} - \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )} - \log {\left (x + \frac {1}{2 b} - \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )}}{b} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=-\frac {\log \left (\frac {2 \, b^{2} x - b - \sqrt {-4 \, a b^{3} + b^{2}}}{2 \, b^{2} x + b - \sqrt {-4 \, a b^{3} + b^{2}}}\right )}{b} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=-\frac {\log \left (\frac {{\left | 2 \, b^{2} x - \sqrt {-4 \, a b + 1} {\left | b \right |} - {\left | b \right |} \right |}}{{\left | 2 \, b^{2} x - \sqrt {-4 \, a b + 1} {\left | b \right |} + {\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]
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Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {1}{a b+\sqrt {b^2-4 a b^3} x-b^2 x^2} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {b^2-4\,a\,b^3}}{\sqrt {b^2}}-\frac {2\,b^2\,x}{\sqrt {b^2}}\right )}{\sqrt {b^2}} \]
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