Optimal. Leaf size=24 \[ -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14}
\begin {gather*} -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x^2+b^2 x^4}{x^2} \, dx &=\int \left (2 a b+\frac {a^2}{x^2}+b^2 x^2\right ) \, dx\\ &=-\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 23, normalized size = 0.96
method | result | size |
default | \(-\frac {a^{2}}{x}+2 a b x +\frac {b^{2} x^{3}}{3}\) | \(23\) |
risch | \(-\frac {a^{2}}{x}+2 a b x +\frac {b^{2} x^{3}}{3}\) | \(23\) |
norman | \(\frac {\frac {1}{3} b^{2} x^{4}+2 a b \,x^{2}-a^{2}}{x}\) | \(26\) |
gosper | \(-\frac {-b^{2} x^{4}-6 a b \,x^{2}+3 a^{2}}{3 x}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 25, normalized size = 1.04 \begin {gather*} \frac {b^{2} x^{4} + 6 \, a b x^{2} - 3 \, a^{2}}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 19, normalized size = 0.79 \begin {gather*} - \frac {a^{2}}{x} + 2 a b x + \frac {b^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 14.26, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 22, normalized size = 0.92 \begin {gather*} \frac {b^2\,x^3}{3}-\frac {a^2}{x}+2\,a\,b\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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