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 = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 270} \[ -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1593
Rubi steps
\begin {align*} \int \left (\frac {a}{x}+b x\right )^2 \, dx &=\int \frac {\left (a+b x^2\right )^2}{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 \[ -\frac {a^2}{x}+2 a b x+\frac {b^2 x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 25, normalized size = 1.04 \[ \frac {b^{2} x^{4} + 6 \, a b x^{2} - 3 \, a^{2}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 22, normalized size = 0.92 \[ \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 0.96 \[ \frac {b^{2} x^{3}}{3}+2 a b x -\frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 22, normalized size = 0.92 \[ \frac {1}{3} \, b^{2} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 22, normalized size = 0.92 \[ \frac {b^2\,x^3}{3}-\frac {a^2}{x}+2\,a\,b\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 19, normalized size = 0.79 \[ - \frac {a^{2}}{x} + 2 a b x + \frac {b^{2} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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