Optimal. Leaf size=190 \[ \frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2} a^{5/4}}+\frac {x}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {193, 321, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{4 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2} a^{5/4}}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 193
Rule 204
Rule 211
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{a+\frac {b}{x^4}} \, dx &=\int \frac {x^4}{b+a x^4} \, dx\\ &=\frac {x}{a}-\frac {b \int \frac {1}{b+a x^4} \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^2}{b+a x^4} \, dx}{2 a}-\frac {\sqrt {b} \int \frac {\sqrt {b}+\sqrt {a} x^2}{b+a x^4} \, dx}{2 a}\\ &=\frac {x}{a}+\frac {\sqrt [4]{b} \int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx}{4 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a^{3/2}}-\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt {b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a^{3/2}}\\ &=\frac {x}{a}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}\\ &=\frac {x}{a}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 173, normalized size = 0.91 \[ \frac {\sqrt {2} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )-\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )+2 \sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )-2 \sqrt {2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )+8 \sqrt [4]{a} x}{8 a^{5/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 119, normalized size = 0.63 \[ -\frac {4 \, a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a^{4} x \left (-\frac {b}{a^{5}}\right )^{\frac {3}{4}} - \sqrt {a^{2} \sqrt {-\frac {b}{a^{5}}} + x^{2}} a^{4} \left (-\frac {b}{a^{5}}\right )^{\frac {3}{4}}}{b}\right ) + a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) - a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (-a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) - 4 \, x}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 172, normalized size = 0.91 \[ \frac {x}{a} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{a}}\right )}{8 \, a^{2}} + \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{a}}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 133, normalized size = 0.70 \[ \frac {x}{a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )}{4 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )}{4 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {b}{a}}}{x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {b}{a}}}\right )}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 179, normalized size = 0.94 \[ -\frac {\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {a} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {a} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {a} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {b}\right )}{a^{\frac {1}{4}}} - \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {a} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {b}\right )}{a^{\frac {1}{4}}}}{8 \, a} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 48, normalized size = 0.25 \[ \frac {x}{a}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x}{{\left (-b\right )}^{1/4}}\right )}{2\,a^{5/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {a^{1/4}\,x}{{\left (-b\right )}^{1/4}}\right )}{2\,a^{5/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 0.12 \[ \operatorname {RootSum} {\left (256 t^{4} a^{5} + b, \left (t \mapsto t \log {\left (- 4 t a + x \right )} \right )\right )} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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