Optimal. Leaf size=68 \[ \frac {(a+b x) \log (x)}{a \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {660, 36, 29, 31}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 660
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*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.46 \begin {gather*} \frac {(a+b x) (\log (x)-\log (a+b x))}{a \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 30, normalized size = 0.44
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{\sqrt {\left (b x +a \right )^{2}}\, a}\) | \(30\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (b x +a \right )}{\left (b x +a \right ) a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (-x \right )}{\left (b x +a \right ) a}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 38, normalized size = 0.56 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.43, size = 16, normalized size = 0.24 \begin {gather*} -\frac {\log \left (b x + a\right ) - \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 10, normalized size = 0.15 \begin {gather*} \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 28, normalized size = 0.41 \begin {gather*} -{\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 46, normalized size = 0.68 \begin {gather*} -\frac {\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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