Optimal. Leaf size=31 \[ \frac {b^2 \log (a x+b)}{a^3}-\frac {b x}{a^2}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.02, antiderivative size = 31, 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 = {263, 43} \[ \frac {b^2 \log (a x+b)}{a^3}-\frac {b x}{a^2}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {x}{a+\frac {b}{x}} \, dx &=\int \frac {x^2}{b+a x} \, dx\\ &=\int \left (-\frac {b}{a^2}+\frac {x}{a}+\frac {b^2}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac {b x}{a^2}+\frac {x^2}{2 a}+\frac {b^2 \log (b+a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.00 \[ \frac {b^2 \log (a x+b)}{a^3}-\frac {b x}{a^2}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 29, normalized size = 0.94 \[ \frac {a^{2} x^{2} - 2 \, a b x + 2 \, b^{2} \log \left (a x + b\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 30, normalized size = 0.97 \[ \frac {b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac {a x^{2} - 2 \, b x}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.97 \[ \frac {x^{2}}{2 a}-\frac {b x}{a^{2}}+\frac {b^{2} \ln \left (a x +b \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 29, normalized size = 0.94 \[ \frac {b^{2} \log \left (a x + b\right )}{a^{3}} + \frac {a x^{2} - 2 \, b x}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.94 \[ \frac {2\,b^2\,\ln \left (b+a\,x\right )+a^2\,x^2-2\,a\,b\,x}{2\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 0.84 \[ \frac {x^{2}}{2 a} - \frac {b x}{a^{2}} + \frac {b^{2} \log {\left (a x + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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