Optimal. Leaf size=17 \[ \frac {b x^2}{2}+\frac {a x^4}{4} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14}
\begin {gather*} \frac {a x^4}{4}+\frac {b x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) x^3 \, dx &=\int \left (b x+a x^3\right ) \, dx\\ &=\frac {b x^2}{2}+\frac {a x^4}{4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^2}{2}+\frac {a x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 15, normalized size = 0.88
method | result | size |
gosper | \(\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}\) | \(15\) |
default | \(\frac {\left (a \,x^{2}+b \right )^{2}}{4 a}\) | \(15\) |
norman | \(\frac {\frac {1}{4} a \,x^{5}+\frac {1}{2} b \,x^{3}}{x}\) | \(18\) |
risch | \(\frac {a \,x^{4}}{4}+\frac {b \,x^{2}}{2}+\frac {b^{2}}{4 a}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {a\,x^4}{4}+\frac {b\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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