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 = 0.132312, antiderivative size = 190, normalized size of antiderivative = 1., 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 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
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*}
Mathematica [A] time = 0.0489001, 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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Maple [A] time = 0.007, size = 133, normalized size = 0.7 \begin{align*}{\frac{x}{a}}-{\frac{\sqrt{2}}{8\,a}\sqrt [4]{{\frac{b}{a}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{b}{a}}}x\sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{b}{a}}}x\sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84611, size = 284, normalized size = 1.49 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.313245, size = 22, normalized size = 0.12 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09517, size = 232, normalized size = 1.22 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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