Optimal. Leaf size=42 \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.0102541, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^4\right )^{3/2}} \, dx &=-\frac{1}{2 a x^2 \sqrt{a+b x^4}}-\frac{(2 b) \int \frac{x}{\left (a+b x^4\right )^{3/2}} \, dx}{a}\\ &=-\frac{1}{2 a x^2 \sqrt{a+b x^4}}-\frac{b x^2}{a^2 \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [A] time = 0.0069317, size = 29, normalized size = 0.69 \[ -\frac{a+2 b x^4}{2 a^2 x^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 0.6 \begin{align*} -{\frac{2\,b{x}^{4}+a}{2\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958572, size = 49, normalized size = 1.17 \begin{align*} -\frac{b x^{2}}{2 \, \sqrt{b x^{4} + a} a^{2}} - \frac{\sqrt{b x^{4} + a}}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48744, size = 78, normalized size = 1.86 \begin{align*} -\frac{{\left (2 \, b x^{4} + a\right )} \sqrt{b x^{4} + a}}{2 \,{\left (a^{2} b x^{6} + a^{3} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01501, size = 46, normalized size = 1.1 \begin{align*} - \frac{1}{2 a \sqrt{b} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{4}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16058, size = 47, normalized size = 1.12 \begin{align*} -\frac{\sqrt{b + \frac{a}{x^{4}}}}{2 \, a^{2}} + \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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