Optimal. Leaf size=69 \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]
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Rubi [A] time = 0.0233859, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {640, 608, 31} \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{\left (2 b^2 d-2 a b e\right ) \int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{\left (\left (2 b^2 d-2 a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{a b+b^2 x} \, dx}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{(b d-a e) (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0172236, size = 40, normalized size = 0.58 \[ \frac{(a+b x) ((b d-a e) \log (a+b x)+b e x)}{b^2 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 45, normalized size = 0.7 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) ae-\ln \left ( bx+a \right ) bd-bxe \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01973, size = 80, normalized size = 1.16 \begin{align*} \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{a \sqrt{\frac{1}{b^{2}}} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48752, size = 54, normalized size = 0.78 \begin{align*} \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.369018, size = 20, normalized size = 0.29 \begin{align*} \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13261, size = 62, normalized size = 0.9 \begin{align*} \frac{x e \mathrm{sgn}\left (b x + a\right )}{b} + \frac{{\left (b d \mathrm{sgn}\left (b x + a\right ) - a e \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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