Optimal. Leaf size=25 \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]
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Rubi [A] time = 0.0040289, antiderivative size = 25, 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, 261} \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 261
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^4}}{\sqrt{1+x^3}} \, dx &=\frac{\sqrt{a x^4} \int \frac{x^2}{\sqrt{1+x^3}} \, dx}{x^2}\\ &=\frac{2 \sqrt{a x^4} \sqrt{1+x^3}}{3 x^2}\\ \end{align*}
Mathematica [A] time = 0.004558, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) }{3\,{x}^{2}}\sqrt{a{x}^{4}}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82366, size = 38, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (\sqrt{a} x^{3} + \sqrt{a}\right )}}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.992094, size = 47, normalized size = 1.88 \begin{align*} \frac{2 \, \sqrt{a x^{4}} \sqrt{x^{3} + 1}}{3 \, x^{2}} \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{a x^{4}}}{\sqrt{\left (x + 1\right ) \left (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 = 1.15782, size = 16, normalized size = 0.64 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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