Optimal. Leaf size=74 \[ \frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}} \]
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Rubi [A] time = 0.0264302, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, 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 &=-\left (-\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}}\right )\\ &=-\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 = 0.01265, size = 33, normalized size = 0.45 \[ \frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{-(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, size = 32, 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 [C] time = 2.07598, size = 161, normalized size = 2.18 \begin{align*} -\sqrt{-\frac{1}{a^{2}}} \log \left (\frac{i \, a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 \, b x + a}{2 \, b}\right ) + \sqrt{-\frac{1}{a^{2}}} \log \left (\frac{-i \, a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 \, b x + a}{2 \, b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- \left (a + b x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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