Optimal. Leaf size=68 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+b x+c x^2}}{c} \]
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Rubi [A] time = 0.0242194, antiderivative size = 68, 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, 621, 206} \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+b x+c x^2}}{c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x}{\sqrt{a+b x+c x^2}} \, dx &=\frac{e \sqrt{a+b x+c x^2}}{c}+\frac{(2 c d-b e) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c}\\ &=\frac{e \sqrt{a+b x+c x^2}}{c}+\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c}\\ &=\frac{e \sqrt{a+b x+c x^2}}{c}+\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0794227, size = 66, normalized size = 0.97 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+x (b+c x)}}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 81, normalized size = 1.2 \begin{align*}{\frac{e}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{be}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{d\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32628, size = 400, normalized size = 5.88 \begin{align*} \left [\frac{4 \, \sqrt{c x^{2} + b x + a} c e -{\left (2 \, c d - b e\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right )}{4 \, c^{2}}, \frac{2 \, \sqrt{c x^{2} + b x + a} c e -{\left (2 \, c d - b e\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{2 \, c^{2}}\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{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10093, size = 88, normalized size = 1.29 \begin{align*} \frac{\sqrt{c x^{2} + b x + a} e}{c} - \frac{{\left (2 \, c d - b e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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