Optimal. Leaf size=25 \[ \frac {2 \sqrt {a x^2+b x^5}}{3 b x} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1602}
\begin {gather*} \frac {2 \sqrt {a x^2+b x^5}}{3 b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a x^2+b x^5}} \, dx &=\frac {2 \sqrt {a x^2+b x^5}}{3 b x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^2 \left (a+b x^3\right )}}{3 b x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 27, normalized size = 1.08
method | result | size |
trager | \(\frac {2 \sqrt {b \,x^{5}+a \,x^{2}}}{3 b x}\) | \(22\) |
gosper | \(\frac {2 x \left (b \,x^{3}+a \right )}{3 b \sqrt {b \,x^{5}+a \,x^{2}}}\) | \(27\) |
default | \(\frac {2 x \left (b \,x^{3}+a \right )}{3 b \sqrt {b \,x^{5}+a \,x^{2}}}\) | \(27\) |
risch | \(\frac {2 x \left (b \,x^{3}+a \right )}{3 \sqrt {x^{2} \left (b \,x^{3}+a \right )}\, b}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 14, normalized size = 0.56 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, \sqrt {b x^{5} + a x^{2}}}{3 \, b x} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 29, normalized size = 1.16 \begin {gather*} -\frac {2 \, \sqrt {a} \mathrm {sgn}\left (x\right )}{3 \, b} + \frac {2 \, \sqrt {b x^{3} + a}}{3 \, b \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.19, size = 21, normalized size = 0.84 \begin {gather*} \frac {2\,\sqrt {b\,x^5+a\,x^2}}{3\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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