Optimal. Leaf size=106 \[ \frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}} \]
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Rubi [A] time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1165, 628} \[ \frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}} \]
Antiderivative was successfully verified.
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Rule 628
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx &=-\frac {\sqrt [4]{b} \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{2 \sqrt {2} \sqrt [4]{a}}\\ &=-\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 91, normalized size = 0.86 \[ \frac {\sqrt [4]{b} \left (\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x-\sqrt {a}-\sqrt {b} x^2\right )\right )}{2 \sqrt {2} \sqrt [4]{a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 151, normalized size = 1.42 \[ \left [\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \log \left (\frac {b x^{4} + 4 \, \sqrt {a} \sqrt {b} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} + a x\right )} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} + a}{b x^{4} + a}\right ), -\sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\sqrt {\frac {1}{2}} x \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}\right ) + \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} - a x\right )} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{a}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 254, normalized size = 2.40 \[ \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \sqrt {a}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \sqrt {a}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \sqrt {a}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.37, size = 70, normalized size = 0.66 \[ \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{4 \, a^{\frac {1}{4}}} - \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{4 \, a^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.76, size = 43, normalized size = 0.41 \[ \frac {\sqrt {2}\,b^{1/4}\,\mathrm {atanh}\left (\frac {2\,\sqrt {2}\,a^{1/4}\,b^{11/4}\,x}{2\,\sqrt {a}\,b^{5/2}+2\,b^3\,x^2}\right )}{2\,a^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 131, normalized size = 1.24 \[ - \frac {\sqrt {2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \log {\left (- \frac {\sqrt {2} \sqrt {a} x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} + \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \log {\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} + \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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