Optimal. Leaf size=23 \[ -\frac {b^2}{3 x^3}-\frac {2 a b}{x}+a^2 x \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {199, 276}
\begin {gather*} a^2 x-\frac {2 a b}{x}-\frac {b^2}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 276
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^2 \, dx &=\int \frac {\left (b+a x^2\right )^2}{x^4} \, dx\\ &=\int \left (a^2+\frac {b^2}{x^4}+\frac {2 a b}{x^2}\right ) \, dx\\ &=-\frac {b^2}{3 x^3}-\frac {2 a b}{x}+a^2 x\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {b^2}{3 x^3}-\frac {2 a b}{x}+a^2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 22, normalized size = 0.96
method | result | size |
default | \(-\frac {b^{2}}{3 x^{3}}-\frac {2 a b}{x}+a^{2} x\) | \(22\) |
risch | \(a^{2} x +\frac {-2 a b \,x^{2}-\frac {1}{3} b^{2}}{x^{3}}\) | \(24\) |
norman | \(\frac {a^{2} x^{4}-2 a b \,x^{2}-\frac {1}{3} b^{2}}{x^{3}}\) | \(25\) |
gosper | \(\frac {3 a^{2} x^{4}-6 a b \,x^{2}-b^{2}}{3 x^{3}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 21, normalized size = 0.91 \begin {gather*} a^{2} x - \frac {2 \, a b}{x} - \frac {b^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 1.13 \begin {gather*} \frac {3 \, a^{2} x^{4} - 6 \, a b x^{2} - b^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.96 \begin {gather*} a^{2} x + \frac {- 6 a b x^{2} - b^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 22, normalized size = 0.96 \begin {gather*} a^{2} x - \frac {6 \, a b x^{2} + b^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 1.04 \begin {gather*} a^2\,x-\frac {\frac {b^2}{3}+2\,a\,b\,x^2}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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