Optimal. Leaf size=68 \[ \frac{\log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0215129, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {646, 36, 29, 31} \[ \frac{\log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{x \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{x} \, dx}{a b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (b \left (a b+b^2 x\right )\right ) \int \frac{1}{a b+b^2 x} \, dx}{a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(a+b x) \log (x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0097142, size = 31, normalized size = 0.46 \[ \frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 30, normalized size = 0.4 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65685, size = 38, normalized size = 0.56 \begin{align*} -\frac{\log \left (b x + a\right ) - \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.532106, size = 10, normalized size = 0.15 \begin{align*} \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27777, size = 38, normalized size = 0.56 \begin{align*} -{\left (\frac{\log \left ({\left | b x + a \right |}\right )}{a} - \frac{\log \left ({\left | x \right |}\right )}{a}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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