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 = 0.03, 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 31
Rule 616
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.86, size = 59, normalized size = 1.90 \[ \frac {\log \left (\frac {2 \, b^{2} x + b + \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right ) - \log \left (\frac {2 \, b^{2} x - b + \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 54, normalized size = 1.74 \[ -\frac {\log \left (\frac {{\left | 2 \, b^{2} x + \sqrt {-4 \, a b + 1} {\left | b \right |} - {\left | b \right |} \right |}}{{\left | 2 \, b^{2} x + \sqrt {-4 \, a b + 1} {\left | b \right |} + {\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 1.00 \[ \frac {2 \arctanh \left (\frac {2 b^{2} x +\sqrt {-\left (4 a b -1\right ) b^{2}}}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 51, normalized size = 1.65 \[ -\frac {\log \left (\frac {2 \, b^{2} x - b + \sqrt {-4 \, a b^{3} + b^{2}}}{2 \, b^{2} x + b + \sqrt {-4 \, a b^{3} + b^{2}}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 38, normalized size = 1.23 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {b^2-4\,a\,b^3}}{\sqrt {b^2}}+\frac {2\,b^2\,x}{\sqrt {b^2}}\right )}{\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 56, normalized size = 1.81 \[ - \frac {\log {\left (x - \frac {1}{2 b} + \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )} - \log {\left (x + \frac {1}{2 b} + \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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