Optimal. Leaf size=25 \[ \frac {2 \sqrt {x^3+1} \sqrt {a x^4}}{3 x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ \frac {2 \sqrt {x^3+1} \sqrt {a x^4}}{3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 19, normalized size = 0.76 \[ \frac {2 \, \sqrt {a x^{4}} \sqrt {x^{3} + 1}}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 12, normalized size = 0.48 \[ \frac {2}{3} \, \sqrt {x^{3} + 1} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.24 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \sqrt {a \,x^{4}}}{3 \sqrt {x^{3}+1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.26, size = 28, normalized size = 1.12 \[ \frac {2 \, {\left (\sqrt {a} x^{3} + \sqrt {a}\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.91, size = 20, normalized size = 0.80 \[ \frac {2\,\sqrt {a}\,\sqrt {x^3+1}\,\sqrt {x^4}}{3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{4}}}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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