Optimal. Leaf size=31 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b^2-4 a b^3}+2 b^2 x}{b}\right )}{b} \]
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Rubi [A] time = 0.0291565, antiderivative size = 58, normalized size of antiderivative = 1.87, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {616, 31} \[ \frac{\log \left (\sqrt{b^2-4 a b^3}+2 b^2 x+b\right )}{b}-\frac{\log \left (-\sqrt{b^2-4 a b^3}-2 b^2 x+b\right )}{b} \]
Antiderivative was successfully verified.
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Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a b-\sqrt{b^2-4 a b^3} x-b^2 x^2} \, dx &=-\left (b \int \frac{1}{\frac{1}{2} \left (-b-\sqrt{b^2-4 a b^3}\right )-b^2 x} \, dx\right )+b \int \frac{1}{\frac{1}{2} \left (b-\sqrt{b^2-4 a b^3}\right )-b^2 x} \, dx\\ &=-\frac{\log \left (b-\sqrt{b^2-4 a b^3}-2 b^2 x\right )}{b}+\frac{\log \left (b+\sqrt{b^2-4 a b^3}+2 b^2 x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0311477, size = 32, normalized size = 1.03 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{-b^2 (4 a b-1)}+2 b^2 x}{b}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.147, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{1}{b}{\it Artanh} \left ({\frac{2\,{b}^{2}x+\sqrt{-{b}^{2} \left ( 4\,ab-1 \right ) }}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22274, size = 82, normalized size = 2.65 \begin{align*} -\frac{\log \left (\frac{2 \, b^{2} x + \sqrt{-4 \, a b^{3} + b^{2}} - \sqrt{b^{2}}}{2 \, b^{2} x + \sqrt{-4 \, a b^{3} + b^{2}} + \sqrt{b^{2}}}\right )}{\sqrt{b^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.3588, size = 128, normalized size = 4.13 \begin{align*} \frac{\log \left (\frac{2 \, b^{2} x + b + \sqrt{-4 \, a b^{3} + b^{2}}}{b}\right ) - \log \left (\frac{2 \, b^{2} x - b + \sqrt{-4 \, a b^{3} + b^{2}}}{b}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.450968, size = 56, normalized size = 1.81 \begin{align*} - \frac{\log{\left (x - \frac{1}{2 b} + \frac{\sqrt{- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )} - \log{\left (x + \frac{1}{2 b} + \frac{\sqrt{- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21375, size = 73, normalized size = 2.35 \begin{align*} -\frac{\log \left (\frac{{\left | 2 \, b^{2} x + \sqrt{-4 \, a b + 1}{\left | b \right |} -{\left | b \right |} \right |}}{{\left | 2 \, b^{2} x + \sqrt{-4 \, a b + 1}{\left | b \right |} +{\left | b \right |} \right |}}\right )}{{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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