Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.0252541, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )}{4 a}\\ &=-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{2 a b}\\ &=-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0572175, size = 69, normalized size = 1.5 \[ \frac{\sqrt{b} \sqrt{\frac{a x^4}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )-\sqrt{a} x^2}{2 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 67, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{2}{a}^{{\frac{3}{2}}}-\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \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 [B] time = 1.50679, size = 365, normalized size = 7.93 \begin{align*} \left [-\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + b\right )} \sqrt{a} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right )}{4 \,{\left (a^{3} x^{4} + a^{2} b\right )}}, -\frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} +{\left (a x^{4} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.99204, size = 187, normalized size = 4.07 \begin{align*} - \frac{2 a^{3} x^{4} \sqrt{1 + \frac{b}{a x^{4}}}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{3} x^{4} \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{4} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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