Optimal. Leaf size=23 \[ -\frac {2 \sqrt {a x+b x^4}}{3 a x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2039}
\begin {gather*} -\frac {2 \sqrt {a x+b x^4}}{3 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x+b x^4}} \, dx &=-\frac {2 \sqrt {a x+b x^4}}{3 a x^2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x \left (a+b x^3\right )}}{3 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {2 \sqrt {b \,x^{4}+a x}}{3 a \,x^{2}}\) | \(20\) |
trager | \(-\frac {2 \sqrt {b \,x^{4}+a x}}{3 a \,x^{2}}\) | \(20\) |
elliptic | \(-\frac {2 \sqrt {b \,x^{4}+a x}}{3 a \,x^{2}}\) | \(20\) |
gosper | \(-\frac {2 \left (b \,x^{3}+a \right )}{3 x a \sqrt {b \,x^{4}+a x}}\) | \(27\) |
risch | \(-\frac {2 \left (b \,x^{3}+a \right )}{3 a x \sqrt {x \left (b \,x^{3}+a \right )}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (b x^{4} + a x\right )}}{3 \, \sqrt {b x^{3} + a} a x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.63, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2 \, \sqrt {b x^{4} + a x}}{3 \, a x^{2}} \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 {1}{x^{2} \sqrt {x \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 = 0.86, size = 14, normalized size = 0.61 \begin {gather*} -\frac {2 \, \sqrt {b + \frac {a}{x^{3}}}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.13, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2\,\sqrt {b\,x^4+a\,x}}{3\,a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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