Optimal. Leaf size=46 \[ \frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
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Rubi [A] time = 0.0266344, 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} \[ \frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{2 a \sqrt{a+b x^4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac{1}{2 a \sqrt{a+b x^4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{2 a b}\\ &=\frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0067697, size = 36, normalized size = 0.78 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right )}{2 a \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 44, normalized size = 1. \begin{align*}{\frac{1}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{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 [A] time = 1.51133, size = 301, normalized size = 6.54 \begin{align*} \left [\frac{{\left (b x^{4} + a\right )} \sqrt{a} \log \left (\frac{b x^{4} - 2 \, \sqrt{b x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + 2 \, \sqrt{b x^{4} + a} a}{4 \,{\left (a^{2} b x^{4} + a^{3}\right )}}, \frac{{\left (b x^{4} + a\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{4} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{4} + a} a}{2 \,{\left (a^{2} b x^{4} + a^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.73768, size = 184, normalized size = 4. \begin{align*} \frac{2 a^{3} \sqrt{1 + \frac{b x^{4}}{a}}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{3} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{2} b x^{4} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{2} b x^{4} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10521, size = 55, normalized size = 1.2 \begin{align*} \frac{\arctan \left (\frac{\sqrt{b x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a} + \frac{1}{2 \, \sqrt{b x^{4} + a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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