Optimal. Leaf size=103 \[ -\frac {a+b x}{a x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b \log (x) (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 44} \[ -\frac {a+b x}{a x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b \log (x) (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{x^2 \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 \left (\frac {1}{a b x^2}-\frac {1}{a^2 x}+\frac {b}{a^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{a x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) \log (x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.40 \[ -\frac {(a+b x) (-b x \log (a+b x)+a+b x \log (x))}{a^2 x \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 26, normalized size = 0.25 \[ \frac {b x \log \left (b x + a\right ) - b x \log \relax (x) - a}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 37, normalized size = 0.36 \[ {\left (\frac {b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {1}{a x}\right )} \mathrm {sgn}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 41, normalized size = 0.40 \[ \frac {\left (b x +a \right ) \left (-b x \ln \relax (x )+b x \ln \left (b x +a \right )-a \right )}{\sqrt {\left (b x +a \right )^{2}}\, a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 65, normalized size = 0.63 \[ \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 68, normalized size = 0.66 \[ \frac {a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a}{\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}\right )}{{\left (a^2\right )}^{3/2}}-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 19, normalized size = 0.18 \[ - \frac {1}{a x} + \frac {b \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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