Integrand size = 27, antiderivative size = 75 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1176, 631, 210} \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}} \]
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Rule 210
Rule 631
Rule 1176
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx \\ & = \frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}} \\ & = -\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt [4]{b} \left (-\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right )}{\sqrt {2} \sqrt [4]{a}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(51)=102\).
Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.72
method | result | size |
default | \(\frac {\sqrt {b}\, \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \sqrt {a}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(204\) |
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Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\left [\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \log \left (\frac {b x^{4} - 4 \, \sqrt {a} \sqrt {b} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} - a x\right )} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} + a}{b x^{4} + a}\right ), \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\right ) + \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} + a x\right )} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}{a}\right )\right ] \]
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Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=- \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (- \frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} \]
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Exception generated. \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\text {Exception raised: TypeError} \]
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Time = 13.90 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt {2}\,b^{1/4}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{3/4}\,x^3}{2\,a^{3/4}}+\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )\right )}{4\,a^{1/4}} \]
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