Optimal. Leaf size=23 \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]
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Rubi [A] time = 0.0043128, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 264} \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{a}{x^7}}}{\sqrt{1+x^5}} \, dx &=\left (\sqrt{\frac{a}{x^7}} x^{7/2}\right ) \int \frac{1}{x^{7/2} \sqrt{1+x^5}} \, dx\\ &=-\frac{2}{5} \sqrt{\frac{a}{x^7}} x \sqrt{1+x^5}\\ \end{align*}
Mathematica [A] time = 0.0038984, size = 23, normalized size = 1. \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 37, normalized size = 1.6 \begin{align*} -{\frac{2\,x \left ( 1+x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}\sqrt{{\frac{a}{{x}^{7}}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55219, size = 55, normalized size = 2.39 \begin{align*} -\frac{2 \,{\left (\sqrt{a} x^{6} + \sqrt{a} x\right )}}{5 \, \sqrt{x^{4} - x^{3} + x^{2} - x + 1} \sqrt{x + 1} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22383, size = 46, normalized size = 2. \begin{align*} -\frac{2}{5} \, \sqrt{x^{5} + 1} x \sqrt{\frac{a}{x^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{7}}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24845, size = 38, normalized size = 1.65 \begin{align*} -\frac{2 \, a^{3}{\left (\frac{\sqrt{a + \frac{a}{x^{5}}}}{a^{2}} - \frac{1}{a^{\frac{3}{2}}}\right )}}{5 \,{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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