Optimal. Leaf size=28 \[ -\frac {b^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {a^2}{x} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 276}
\begin {gather*} -\frac {a^2}{x}-\frac {2 a b}{3 x^3}-\frac {b^2}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 269
Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^2} \, dx &=\int \frac {\left (b+a x^2\right )^2}{x^6} \, dx\\ &=\int \left (\frac {b^2}{x^6}+\frac {2 a b}{x^4}+\frac {a^2}{x^2}\right ) \, dx\\ &=-\frac {b^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {a^2}{x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {b^2}{5 x^5}-\frac {2 a b}{3 x^3}-\frac {a^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 25, normalized size = 0.89
method | result | size |
default | \(-\frac {b^{2}}{5 x^{5}}-\frac {2 a b}{3 x^{3}}-\frac {a^{2}}{x}\) | \(25\) |
norman | \(\frac {-a^{2} x^{4}-\frac {2}{3} a b \,x^{2}-\frac {1}{5} b^{2}}{x^{5}}\) | \(26\) |
risch | \(\frac {-a^{2} x^{4}-\frac {2}{3} a b \,x^{2}-\frac {1}{5} b^{2}}{x^{5}}\) | \(26\) |
gosper | \(-\frac {15 a^{2} x^{4}+10 a b \,x^{2}+3 b^{2}}{15 x^{5}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 27, normalized size = 0.96 \begin {gather*} \frac {- 15 a^{2} x^{4} - 10 a b x^{2} - 3 b^{2}}{15 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 26, normalized size = 0.93 \begin {gather*} -\frac {15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} -\frac {a^2\,x^4+\frac {2\,a\,b\,x^2}{3}+\frac {b^2}{5}}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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