Optimal. Leaf size=18 \[ -\frac {a}{x}+b x+\frac {c x^3}{3} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14}
\begin {gather*} -\frac {a}{x}+b x+\frac {c x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^2} \, dx &=\int \left (b+\frac {a}{x^2}+c x^2\right ) \, dx\\ &=-\frac {a}{x}+b x+\frac {c x^3}{3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {a}{x}+b x+\frac {c x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.94
method | result | size |
default | \(-\frac {a}{x}+b x +\frac {c \,x^{3}}{3}\) | \(17\) |
risch | \(-\frac {a}{x}+b x +\frac {c \,x^{3}}{3}\) | \(17\) |
norman | \(\frac {\frac {1}{3} c \,x^{4}+b \,x^{2}-a}{x}\) | \(20\) |
gosper | \(-\frac {-c \,x^{4}-3 b \,x^{2}+3 a}{3 x}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 20, normalized size = 1.11 \begin {gather*} \frac {c x^{4} + 3 \, b x^{2} - 3 \, a}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.67 \begin {gather*} - \frac {a}{x} + b x + \frac {c x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.17, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c x^{3} + b x - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 16, normalized size = 0.89 \begin {gather*} b\,x-\frac {a}{x}+\frac {c\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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