Optimal. Leaf size=28 \[ \frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 29}
\begin {gather*} \frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b x\right )}{b \sqrt {c (a+b x)^2}}\\ &=\frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 27, normalized size = 0.96
method | result | size |
default | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{b \sqrt {c \left (b x +a \right )^{2}}}\) | \(27\) |
risch | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{b \sqrt {c \left (b x +a \right )^{2}}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.54 \begin {gather*} \frac {\log \left (x + \frac {a}{b}\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 42, normalized size = 1.50 \begin {gather*} \frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (b x + a\right )}{b^{2} c x + a b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 34, normalized size = 1.21 \begin {gather*} \frac {\log \left ({\left | b x + a \right |} \sqrt {{\left | c \right |}} {\left | \mathrm {sgn}\left (b x + a\right ) \right |}\right )}{b \sqrt {c} \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 = 1.33, size = 29, normalized size = 1.04 \begin {gather*} \frac {\ln \left (\sqrt {c}\,\left (a+b\,x\right )+\sqrt {c\,{\left (a+b\,x\right )}^2}\right )}{b\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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