Optimal. Leaf size=18 \[ -\frac {a}{3 x^3}-\frac {b}{x}+c x \]
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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}{3 x^3}-\frac {b}{x}+c x \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^4} \, dx &=\int \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right ) \, dx\\ &=-\frac {a}{3 x^3}-\frac {b}{x}+c x\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {a}{3 x^3}-\frac {b}{x}+c x \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 | \(-\frac {a}{3 x^{3}}-\frac {b}{x}+c x\) | \(17\) |
risch | \(c x +\frac {-b \,x^{2}-\frac {a}{3}}{x^{3}}\) | \(19\) |
gosper | \(-\frac {-3 c \,x^{4}+3 b \,x^{2}+a}{3 x^{3}}\) | \(20\) |
norman | \(\frac {c \,x^{4}-b \,x^{2}-\frac {1}{3} a}{x^{3}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 17, normalized size = 0.94 \begin {gather*} c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 21, normalized size = 1.17 \begin {gather*} \frac {3 \, c x^{4} - 3 \, b x^{2} - a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.94 \begin {gather*} c x + \frac {- a - 3 b x^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.64, size = 17, normalized size = 0.94 \begin {gather*} c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} c\,x-\frac {b\,x^2+\frac {a}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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