Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^4}}{4 a x^4} \]
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Rubi [A] time = 0.0259341, antiderivative size = 50, 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} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x^4}}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{a+b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{a+b x^4}}{4 a x^4}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )}{8 a}\\ &=-\frac{\sqrt{a+b x^4}}{4 a x^4}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )}{4 a}\\ &=-\frac{\sqrt{a+b x^4}}{4 a x^4}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0550041, size = 64, normalized size = 1.28 \[ \frac{b \sqrt{a+b x^4} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^4}{a}+1}\right )}{2 \sqrt{\frac{b x^4}{a}+1}}-\frac{a}{2 b x^4}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 48, normalized size = 1. \begin{align*} -{\frac{1}{4\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{b}{4}\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 = 1.48439, size = 265, normalized size = 5.3 \begin{align*} \left [\frac{\sqrt{a} b x^{4} \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}{8 \, a^{2} x^{4}}, -\frac{\sqrt{-a} b x^{4} \arctan \left (\frac{\sqrt{b x^{4} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{4} + a} a}{4 \, a^{2} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.46112, size = 46, normalized size = 0.92 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} + 1}}{4 a x^{2}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11564, size = 65, normalized size = 1.3 \begin{align*} -\frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{4} + a}}{a b x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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