Optimal. Leaf size=56 \[ \frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9} \]
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Rubi [A] time = 0.0156218, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {640, 608, 31} \[ \frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9} \]
Antiderivative was successfully verified.
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Rule 640
Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{\sqrt{9+12 x+4 x^2}} \, dx &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{1}{2} (2 d-3 e) \int \frac{1}{\sqrt{9+12 x+4 x^2}} \, dx\\ &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{((2 d-3 e) (6+4 x)) \int \frac{1}{6+4 x} \, dx}{2 \sqrt{9+12 x+4 x^2}}\\ &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt{9+12 x+4 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0153117, size = 42, normalized size = 0.75 \[ \frac{(2 x+3) ((2 d-3 e) \log (2 x+3)+e (2 x+3))}{4 \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( 3+2\,x \right ) \left ( 2\,\ln \left ( 3+2\,x \right ) d-3\,e\ln \left ( 3+2\,x \right ) +2\,ex \right ) }{4}{\frac{1}{\sqrt{ \left ( 3+2\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55833, size = 41, normalized size = 0.73 \begin{align*} \frac{1}{2} \, d \log \left (x + \frac{3}{2}\right ) - \frac{3}{4} \, e \log \left (x + \frac{3}{2}\right ) + \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60149, size = 54, normalized size = 0.96 \begin{align*} \frac{1}{2} \, e x + \frac{1}{4} \,{\left (2 \, d - 3 \, e\right )} \log \left (2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\sqrt{\left (2 x + 3\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1582, size = 62, normalized size = 1.11 \begin{align*} -\frac{1}{4} \,{\left (2 \, d - 3 \, e\right )} \log \left ({\left | -2 \, x + \sqrt{4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) + \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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